I also want to remind you, that it is respectless and rude to NOT take the shoes off when entering the dojo, especially in Japan,Korea and China. You will find plenty of martial art schools in for example tokyo, which are able to teach you PROPER self-defense while wearing no shoes.
This discussion is totally platitudinous.
Shoes are traditionally worn in Chinese Martial Arts. Along with whatever clothes you wear normally on the street. Doesn't seem to hinder us at all.
I primarily practice TCMA and FMA, and I prefer to wear my regular clothes. That means jeans, whatever shirt I'm wearing that day, and shoes. No need for sports wear, even. I'll even train in a coat sometimes if I'm outside and it's cold - makes chisau a bit interesting! I've even been known to "forget" to bring my gi to aikido practice so I can train in my regular clothes on occasion
I do think it's a good idea to train in shoes, and on different surfaces outside of the dojo. Have you ever practiced your footwork on rough surfaces, with roots or curbs that you might trip over, and changes in height? You'll find it's quite different from training on a perfectly flat, smooth surface. Moreover, you may find that when you pivot on your feet, it can be quite a bit more difficult if you have rubber-soled shoes and are on a very grippy surface. Likewise, there are surfaces where your feet can slide, or dig into the ground. If your first time experiencing any of that is defending yourself on the street, you may very well find yourself on the ground.
I should also state that it's generally rude to wear shoes in Japan and Korea only because that's a cultural norm everywhere. For one, it keeps the floors clean, and secondly, traditional tatami can be damaged if you walk on them with shoes all of the time. So, the reasons, as you know, are quite practical. If this isn't a practical concern for your dojo, there's no need to worry about it. If it is, then go ahead and take your shoes off. I don't see the issue, really.