But electrons don't, and so we have quantum mechanics, i.e. have a set of mathematical equations that predict exactly what electrons will do, in some case exactly and in other cases probabilistically, but we still don't really know why the electron doesn't collapse into the nucleus.
The QM explanation is that electrons can only lose quantized amounts of energy, i.e. they can't just slowly run down. So the electron stays in orbit until something happens such that it can gain a quanta of energy and get thrown out of orbit, or into a higher orbit, or lose a quanta of energy and go into a lower orbit.
Of course in QM electrons don't necessarily exist at one point in space and so don't really orbit nuclei in the same way that planets orbit the sun.
So, yes we have the math to predict what will happen, but we still have no idea why what happens actually does.
“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.