"But the simple reality is they don't teach moveable-do because it's too mind-twisting"
Anyone with an ear does this (without the notation) all the time. I think its something to do with pythagorean Commers. or maybe Dodges.
But you can't do it on a fixed tuned (keyboard) or fretted (where at least it's very difficult) instrument, although singers and fiddlers do it all the time.
Doing it in practice isn't hard (for some), but the theory I agree is impossible. and the only instruction I've ever had regarding it is "make that leading note (that is Te (or Si)) a bit sharper".