“But a circling charged object would emit radiation and lose energy. Over time its orbit would get smaller and smaller. Eventually it would crash into the nucleus.”
I think the key, as you said, is that the electron really isn’t “orbiting” the nucleus, in the classical sense. We just think about it that way because it’s easier for us to conceive than what is really going on. The electrons form standing waves, so they are oscillating in place, rather than moving through the field, so there is no induction. I think what we see as the electron’s position is just the most prominent node of the wave at that particular time.
The particles in the nucleus are standing waves also, and this is the reason that we see quantization, because waves can only form stable arrangements with each other based on the laws of harmonics, so you end up with only rational values being feasible.
I haven't kept up to see if this is still a going theory. It would be interesting if it turned out he was right, but that physicists resisted his theory because the math would be too difficult to work with. After all, points are easier to model than toroidal sheets.