My coach will sometimes go over the lesson afterwards with a few of us, and one thing he often brings up is that he sees some good students who can do even better. He often corrects them so that they can achieve even greater levels of excellence. It might seem to some to be nitpicking, but the truth is, they can't do it with someone who won't bother. And I'm sure this is done in other sports as well, eg in track so you can run faster, or in football so you can do better.
Well, I have tried practicing with shoes, on my own outside the dojang. And probably the real reason we don't train with shoes on is that your shoes will fly off when you kick. LOL. Yes, there are proper shoes that won't have this problem, but it's not likely that the students will want to buy them, especially parents of young kids who outgrow them very quickly, or those who have more than one child learning tkd. I figure it's for reasons of practicality. ^_^