"Shoulds" in philosophy are a tough sell. And in particular, the idea that philosophers "should not" try to overturn ordinary notions is one that's regularly challenged by philosophers. For example: many philosophers have argued that there is no such thing as a "self." Some philosophers have argued that the ordinary notion of belief is incoherent. And challenges to the idea of time, to take your example, have come from within philosophy itself; McTaggart's famous article "The Unreality of Time" offered purely philosophical arguments for abandoning our familiar ways of thinking about time.

It would be not just hard but perverse to argue that philosophy should never challenge our ordinary conceptions -- even if the challenge runs very deep. After all, sometimes we are confused, and even when we're not, there's often something to be learned from meeting the challenge. That said, some attacks on ordianry notions may take those notions to carry more baggage than they really do. The case of the "self" may be a good example. Ordinary people seldom use phrases like "the self," though they do use words like "myself" and "yourself," and while people may say peculiar things about "the self" when pressed by philosophers, it's not clear that the extra stuff plays much of a role in their day-to-day thinking.

Poetry can certainly be used to express profound ideas and attitudes concerning (for want of a better expression) 'the human condition'. These ideas can affect the reader's soul in a powerful way, helped along by the captivating power of the medium itself. And examples of poetry that might be regarded as 'philosophical' in this sense are innumerable. Indeed, one might make a case for claiming that it's the norm rather than the exception, and that this is the primary aspiration of most of the greatest poetry in history, from Homer to Dante to Sylvia Plath.

But does this really count as philosophy? For some people, this is precisely what the best and most important kind of philosophy consists in. For others, however, and particularly within English-speaking academia, philosophy is more a matter of highly technical and abstract theories about the structure of reality, the nature of cognition, and things of that sort. And yet, as it turns out, those kinds of theories have been explored in verse form too. One might compare this with the way in which philosophers from Plato to Berkeley to David Lewis have opted to present their ideas and arguments in the form of witty dialogues. Either approach brings, among other potential advantages, that of simply engaging the reader more effectively than yet another dry prose treatise might.

For an example, consider Lucretius's De Rerum Natura, an epic poem in six books from the first century BC. Lucretius set out and argued for, among other things: an atomistic physics, a plurality of worlds and extraterrestrial life, a theory of natural selection, a materialist account of the soul, a vigorous critique of religion, an account of the origins of human society out of a state of nature, a study of meteorology, and, believe it or not, a discussion of sexual positions. And all in Latin verse. Admittedly, that was then and this is now. Professional philosophers and scientists tend not to express themselves in verse any more -- I can't think of any recent examples (though perhaps others might). But, even into the early modern period, they were still doing so. See, for instance, the book-length philosophical poems, Nosce Teipsum by Sir John Davies (1599) or Psychodia Platonica by Henry More (1642). Some of the scansion and rhyme might have been a bit dodgy, but these authors did nevertheless feel that verse was an appropriate medium for the expression of serious metaphysical and epistemological theories and arguments. There's certainly no incompatibility between the poetic medium and even the most technical kind of philosophy.

Just a footnote to Marc Lange's response (which seems spot on to me). It is worth adding that in serious analytical philosophy there is actually a good deal more agreement on arguments than there might appear to be at first sight, and there is a good deal of pretty secure knowledge.

For what often emerges from the to and fro of debate is essentially something of the form "If you accept A, B and C, then you'd better accept D too". Then one party might endorse A, B and C and conclude that D; and another party might think D is unacceptable, and conclude that one of A, B, or C must be wrong. And another party again (me, often!) might not know how to respond. [A trite example. If you accept act utilitarianism plus some other things, it seems that you should sanction the sheriff hanging an innocent man if that is the way to stop a riot in which more innocent people are killed. Some bite the bullett, some think so much the worse for utilitarianism.]

Now, there may indeed be a loud disagreement between the first two parties. But a lot of hard thinking may have gone into working out what they agree on, namely that "If you accept A, B and C, then you'd better accept D too". [For example, initially people might have thought, wrongly, that accepting A and B alone forced conclusion D, and it took some subtle argument to show that C was playing a role too. Working out what act utilitarianism really does sanction is like this.] And finding that sort of connection can represent a solid achievement of which we can be tolerably certain, even while there remains vigorous disagreement about what to do with the discovery.

Well, why should an "average" person (whoever she is) care about the history of Venetian convents in the sixteenth century, or about the genetics of mice, or about large cardinal axioms in set theory, or the geology of the Caucasus mountains, or Italian linguistics?

No special reason!

Why should it be any different for philosophy? Why indeed should the man in the street care about the limits of a priori knowledge, or about whether we should be structuralists about the natural numbers, or what the correct theory of conditionals is, or whether the theory-theory is better than the simulation theory about how we ascribe mental contents, or ...

Again, no special reason at all.

Of course, philosophers' in-house questions can have their roots in "what is ..?/how is it possible that...?/can we know whether ...?/what should we do ...?" questions of broader appeal. But then in-house questions from historians and scientists, say, ultimately have similar roots. And there is no particular reason why, even if someone has more than their share of general human curiosity, that they should end up interested in one bunch of questions, at one level of abstraction, rather than another bunch at a different level of abstraction. A few like a bit of philosophical sweep to their questions; others (and they are more common) like to get their hands dirty in the nitty gritty of the laboratory or delving in the archives. And lots more aren't interested either way. Why should they be?

The exceptional case, perhaps, is some questions in morals and politics, hard questions that face us all (and which perhaps are uncomfortable to think about). Perhaps more of us should be thinking a bit harder about some of them. Though even here it is open to doubt whether philosophy is especially where to look for the answers to the pressing questions -- it is just part of (sometimes a small part of) the ongoing conversation.

Does there need to be a single, particular contribution that philosophical research makes and other disciplines fail ito make? Of course, science and math clarify concepts and contribute to making empirical predictions. Philosophical research does all of that, too, from time to time. I don't think there needs to be an interesting answer to "What does philosophy do?" that distinguishes philosophy from science and math. All are in pursuit of truth. Philosophers, scientists, and mathematicians are trained somewhat differently, often have somewhat different tools in their toolkits, and come out of somewhat (though overlapping) traditions and so will generally be familiar with different argumentative moves. But these may be differences in degree, not in kind.

The kinds of reasons that are given for favoring one scientific theory over its rivals are a good deal more subtle than "observation is king." To begin with, a theory need not be justly rejected merely because it conflicts with a given observation; sometimes, the observation is appropriately doubted, and sometimes, a given theory is rationally retained despite its failure to fit our observations because blame for the mismatch is placed on other theories ("auxiliary hypotheses") that were used to bring the theory to bear on those observations. (The Copernican model of the solar system, for instance, was retained despite 300 years of failure to observe the stellar parallax it apparently predicts.) By the same token, a theory that fits our observations very well may nevertheless be justly and emphatically rejected on the grounds that it is ad hoc, fails to fit nicely with our other theories, lacks unity or fruitfulness or explanatory power, etc.

Once these familiar features of scientific practice are recognized, then I think the choice among theories in philosophy seems not so dissimilar to the choice among competing scientific theories. Admittedly, much of philosophy is a priori, unlike science. However, the virtues of elegance, parsimony, unity, coherence, explanatory power, and so forth play significant roles in both scientific and philosophical theory-choice.

Finally, philosophy has long been and is increasingly brought into contact with empirical results. A philosophical analysis of causal relations, for instance, that fails to do justice to modern physics has a severe deficiency. Of course, philosophers will disagree about the extent to which modern physics (or even classical physics) discovers causal relations, as well as disagreeing about what physics has revealed about them. Nevertheless, philosophical theorizing hardly takes place in isolation from empirical results. Recent literature in the philosophy of mind and perception offers a host of tremendously potent examples of this.

One thing that can happen is this. Someone asks an inchoate, perhaps rather muddled, question. A respondent -- operating in the approved housestyle of analytical philosophy -- disentangles the issues, and having separated out a crisply formulated question or two, responds briskly to them with clinical precision (well, we try!). And, it can be tempting to think, something important is lost in the process. What lay behind the question -- the depths, so to speak -- are somehow being ignored, and the response doesn't really address the posed worry. So it can seem that the questioner is being fobbed off with BS. Maybe that is your feeling?

But one of the hard lessons when you start philosophy is the realization that behind one's inchoate and confused half-formed questions is often just ... confusion. It is usually not so much a question of hidden depths as muddy shallows. And it can take some time before the penny drops that the analytical philosopher's answers are disentangling the real issues. Are not BS, in other words, but a proper way of resolving muddle.

I can't say for sure how typical your experience is, but I can say for sure that I shared it. Starting to study philosophy is a lot like falling in love (which makes sense, given the literal meaning of the word, right?). When a person falls in love, it's normal to be so enthralled with the beloved that nothing and no one else seems worth a thought. Every conversation eventually turns back toward the beloved. Time spent apart is considered wasted. Other, ordinary folk throw up their hands in exasperation -- and the lover cares naught.

Just like a person in love, whether this condition persists, worsens or improves is up to you. When I was writing my doctoral dissertation, at the same time as birthing, raising and educating eight children, I suffered greatly from what I called "Mommy-Scholar Schizophrenia." It was so difficult, so painful, to shift gears between ethical theory and Disney coloring books! Finally, I came to see that one of the magnificent things about philosophy is its ability to operate on whatever is placed before it. Even Disney coloring books.

Living in an intellectual ivory tower certainly has its appeal, but the mundane isn't going away any time soon. So you can fight it, ignore it...or enjoy it in a special way because you are a philosopher.

My advice: Try not to become frustrated with the simplistic masses; instead, be alert to opportunities for introducing them (gently, tactfully) to this great love you've found. Unlike a beloved person, there's plenty philosophy to go around.

What I'd "make of the idea that philosophers are trying to understand the world through natural language, and not through enough of an observational component, and method" is that this generalization is just plain false. Philosophers throughout history have drawn upon science, mathematics, literature, art, theology, and whatever else they can get their hands on in an attempt to deal with the questions that perplex them.

Once philosophers have achieved enough of an understanding of a question to make it susceptible to serious scientific investigation, what happens is, well, that the question starts to receive serious scientific investigation, and at that point its no longer a question for philosophers but a question for some often new branch of empirical science. This has happened time and again throughout history. So philosophers often aren't in the business of answering questions but rather in the business of clarifying questions: clarifying them enough that empirical work can then be done.

As a result, philosophers at any given time are always worrying about the questions that still need such work to be done on them: the ones that aren't yet ready for scientific study, and perhaps may never be ready for scientific study. Some of the questions that trouble us may not even have answers we could understand. That's possible.

Actually, the presumption here is wrong. It isn't the case that "much of philosophy is concerned with providing a rigorous foundation to truths which are otherwise intuitive and uncontroversial". In particular, that isn't the case in the philosophy of mathematics.

Of course, famously, Frege tried to show that the basic laws of arithmetic (and hence the proposition 1+1 = 2) can be derived from the laws of logic plus definitions. But he did this in order to defend the claim that arithmetical truths are analytic, true in virtue of logic alone, and so explain why those truths are necessarily true and why they necessarily apply to everything. He didn't claim that, prior to his attempted derivation of 1+1 = 2 from pure logic, no one knew it to be true. Rather we weren't in a good position to see clearly the sort of truth that it is, analytic according to Frege.

Unfortunately, one of Frege's putative laws of logic turned out to lead to contradiction, and his foundational edifice crumbled (though neo-Fregeans think that much can be rescued). In part as a response, Russell and Whitehead also famously tried to show that the basic laws of arithmetic can be derived from a small number of more basic laws. But again, it wasn't that they thought that, prior to deriving 1+1 = 2, there is a serious question mark over its truth. In fact, for them, it is the other way about: the rationale for accepting some of the laws of their basic "type theory" is regressive: that is to say, we are to accept their "foundational" laws here rather as we accept laws in physics -- i.e. they generate consequences which we already know to be true.

These days, students are taught how a vast amount of mathematics can be regimented in ZFC (Zermelo Fraenkel set theory with the Axiom of Choice), and in particular are told how to derive proxies for the Peano Axioms for arithmetic, and hence for 1+1 = 2, inside ZFC. One point of this regimenting project is that it gives us a way of calibrating the infinitary commitments of different areas of mathematics by measuring different bits of mathematics against a common yardstick. But again, it isn't as if set theorists think that the basic principles of ZFC are more secure than those of simple arithmetic. On the contrary.

Frege, Russell and Whitehead, and modern set theorists all knew/know that 1+1=2 in the same way we all do. But what that involves is, indeed, another question.