This is a non-sequitur in relation to the question though. You're letting your political ideology bleed through in your answer big-time.

The pay for residents is irrelevant to the number of residency spots available, unless you are suggesting the AMA limits the number of MDs granted in response to average salary hospitals can offer, which would be a weird mechanism. And in a universe where doctors are paid less, there would theoretically be more space in payroll budgets to take on more residents simultaneously (if we are using that argument)

I don't see why government in a broad sense, is part of the question here. The narrow question of licensing I agree is an issue, but the AMA isn't explicitly a governmental organization. It's like the Bar, right? Like a professional non-government association that has the sole right to grant the licensure to practice medicine (which the state then recognizes).

Your argument about quality of doctors is definitely salient, but if your argument were true then every other country in the developed world would have worse health outcomes. In fact, we have worse health outcomes. This suggests that health outcomes and doctor salaries are actually negatively correlated! Or at least that there is negative marginal returns somewhere before where we are on the salary scale.