The most compelling reason not to eat meat is not because it involves killing animals but because it so often involves causing animals to suffer, especially in factory farming. From this point of view there might be nothing wrong with eating free range chicken (because their lives are not miserable) or shrimp (if they are incapable of feeling pain). But as Peter Singer argues, causing unecessary suffering seems wrong. If some factory farming were the only way for some people to get enough iron, then it might be a tough call. But I doubt it is, and if I am wrong about this the argument still applies to many of us who do have easy alternatives. And causing unecessary suffering seems wrong even if the perpetrators are acting perfectly naturally.

There are at least three different kinds of argument in favor of vegetarianism, and each of the arguments have slightly different implications for what is OK.

One argument is concerned with human health (so is more prudential than moral). The idea is that eating dead animals is not healthy for humans, or at least a balanced vegetarian diet is more healthy. This view is not really compatible with eating roadkill, but would be compatible with eating meat if there was insufficient vegetarian food to keep one healthy.

Another argument is concerned with the environment. The idea is that factory farming wastes precious resources (like water) and is inefficient in producing the nutrients humans need. (For details, see "Environmental vegetarianism" in Wikipedia.) This argument also doesn't preclude killing or eating animals where the practices used to raise them are environmentally sound (but it can be developed into a case for a qualified veganism, given the parallel concerns about the environmental impact of the dairy industry).

The "animal rights" argument is concerned with the morality of killing, or causing unnecessary pain to, an animal. Singer, as Prof. Lipton has mentioned, is primarily concerned with pain and suffering. But many vegetarians are opposed even to painless killing of animals for food, sport, or fashion. One need not be religious to believe that if something is sentient then it is something of value and/or it has interests that we ought to respect. To destroy something of value unnecessarily is wrong. To act in a way that unnecessarily violates the interests of another is wrong. These are familiar enough ideas, and some vegetarians just make the added assumption that animals are sentient. As you point out, these ideas don't yield absolute prohibitions against ever eating meat, but they do suggest that routine meat-eating (hunting and leather-wearing) is wrong.