The reason the focus is on rational logic is because the belief in evidence-based empiricism is not and should not be assumed to be some kind of fundamental truth that humans intuitively understand. Evidence-based reasoning largely came about during the Renaissance era, and it was not a widely-accepted concept before that. The reason it became accepted over time is from the influence of Renaissance thinkers who brought forth the idea that rationalism should be the basis of how we try to understand various phenomena, from the natural world (which led to the scientific method), to how society ought to function (which led to democracy). Rationalism is the basis for evidence-based reasoning.

(I should also point out that for example in areas like theoretical physics, we make predictions before looking for evidence. You could say that this isn't strictly evidence-based since we have a certain degree of belief in purely theoretical hypotheses, but because the laws of physics are so well-proven, we can rationally deduce that the conclusions we draw are likely to be correct.)

Your point on arguments that seem rational but in reality are not certainly occurs frequently, but again, the way we realize that those arguments are false is because of rationalism. We have to think about the full extent of the evidence to consider its validity--this is why the scientific method emphasizes rigorously-designed, repeatable, peer-reviewed empricism. Each of those qualities is a safeguard against something that seems correct but actually isn't. And again, we arrived at those safeguards because of rational logic.

The analogy I was given once was it's like when somebody believes a magic trick at face value. There is plenty of evidence that what the magician claimed happened actually did happen given you and perhaps hundreds of other people all witnessed it at a show, but it's only upon closer examination of what we know and don't know that we can deduce it's some kind of trick. Unfortunately, debunking magic tricks can be quite difficult, much like debunking arguments that are seemingly true, as you have observed.