First, there isn't much logical thinking involved in veganism or vegetarianism. At their best, they're about ethics (which I certainly respect). At their worst, they make some very dubious claims about te economics of food and the environment.
But either way, note that people make significant and probably burdensome changes to their lives when it comes to upholding morals pertaining to food. I think that more (liberal) people would go a longer way making actual life changes when it comes to food than just about anything else. They might explain that the things you actually put into your body are the things you should care about most. Only there's little evidence to suggest that's the case. It is much more probable that national policies like increasing minimum wage or reforming education would have a bigger (though indirect) effect on your personal well being than optimizing your diet (as long as it's more or less "normal").
All religions, in this case the healthy diet religion, must not rely on facts in order to gain followers; they must rely on gut feelings, because gut feelings are much more powerful than proof when it comes to motivating life changes. And we just naturally feel that ingested food has to have a major effect on our health and general well being. It just makes sense, right? But, at least in the current state of scientific knowledge on the subject, that intuition is just superstition. After decades of research there's little consensus even on what makes us fat, let alone more subtle or indirect effects diet may have on our health. In fact, it's very probable that even if one day we do know exactly what's healthy to eat and what isn't, and at what quantities, the actual effect-size on our health will be quite small. It's quite possible that within some sane bounds, what food we eat simply does not determine our health to any large degree.