I have a few things to say in response to your question.

First of all, about whether too high a value is placed on rationality in “Western” culture: I feel that rationality is too little valued in the United States at the moment, and that irrationality is celebrated. An extremely popular trope in US books, movies, and television shows is the heroism of a person who “believes” – that is, who accepts on faith something that flies in the face of all evidence and logic. The skeptic, the “man (usually) of science,” is always shown to be wrong, often disastrously so. And many people report with great pride that they hold their particular religious or political beliefs on the basis of no evidence or reason at all. “It’s just what I believe.”

Secondly, as to the nature of our cognitive processes. You’re raising a perfectly sensible empirical question – what are the neurological processes that account for the phenomena we call “thought”? It’s clear that neural networks are responsible, in some way and at some level, for all of our cognition – the mind is made of the brain, and the brain is made of layers of neurons. But that’s not to say that the architecture – the functional organization – of a neural net has to be the architecture of the mind as a whole. Neural networks can implement classical computer architecture, just as molecules can implement complex biological structure. (And by the way, the reverse is true, as well. Lots of research on neural networks is done by using classical computers to simulate/implement connectionist architectures.) Many philosophers think that there are kinds of cognitive processes that require a symbols-and-rules architecture, so that neural networks can’t be the whole story. Logical inference is one of those; we may not engage in it that often, but when we do, it looks like our thought transitions are sensitive to the form of our thoughts, and connectionist architectures are not sensitive to form.

Third, don’t confuse the question of whether there is unconscious cognitive processing with the question of what the architecture of the mind is. The two are independent. Almost everyone with any expertise in cognitive science now accepts that there is a great deal of unconscious information processing going on – but it’s still an open question whether all this processing is connectionist or symbolic.

With respect to points two and three, I suggest you take a look at a book by cognitive scientist Gary Marcus called The Algebraic Mind.

Finally – intuitions: philosophers call something an intuition when it really, really, really, REALLY seems to you to be true, but you can’t give any reasons why you think it’s true. Where intuitions come from, in that sense, is anybody’s guess. Some of them probably do come from unconscious information processing (like when you fall victim to a visual illusion), some may come from our detecting or responding to the operation of cognitive processes (like when you have intuitions of logical relations), some may be memories of things you were told as a child, some may be brain farts. I personally think that philosophy is the business of trying to reconcile everything you think is true with everything else you think is true, so intuitions may as well go into the mix along with everything else. Also like everything else we start out believing, intuitions may have to be abandoned if the pressure gets too great.

Whatever "intuitions" are, however, they can only be the foundation of knowledge of any kind if one can show that they justify the beliefs they give rise to, and to do that, one has to say something about why intuitions are trustworthy. Descartes's foundational intuition did well on that score -- as he said, in the Second Meditation: "So that it must...be maintained, all things being maturely and carefully considered, that this proposition I am, I exist, is necessarily true each time it is expressed by me, or conceived in my mind." (Whether he could derive from that one intuition everything else that he thought he could, with equal certainty is very dubious, and most commentators concede that he could not. ) Apart from Descartes's cogito, though, the intuitions to which philosophers appeal are mostly subject to doubt, and rarely command universal consent. Since the status of intuitions is open to question, the beliefs and theories they support are open to question as well.

On the other hand, intuitions are often all we've got.