Work is work ... and there's nothing wrong wearing a pair of Carharts to teach. Indeed, a little bit of dirt, or wear, helps your 'credibility.'
I can't speak for others, but I consider 'leadership' an essential part of teaching. You can't push a rope- you have to make them WANT to learn. The first requirement is for you to convince them that you're worth listening to, that this matter is worth learning. "Dressing the part" is a requirement. You're speaking as 'senior tradesman' to 'eager minds full of mush.'
And senior tradesmen don't wear tuxedos. Or dresses. Or flip-flops. At least, not when they're up a ladder, hanging lights.