While this is true, the real bottleneck is the availability of training positions. Other commenters have pointed out that these positions are generally increasing in number, but not nearly at a rate large enough to make up for the shortfall.

One positive argument to having a body like the AMA leverage some control over medical schools is that it helps to ensure that schools only open/expand if the AMA thinks that their students will have reasonably good chances of getting residency positions. It helps to avoid the sort of situation that is currently going on in law school and PhD programs where there are no where near enough positions available for all the students who are graduating from those programs.

And I do generally agree with you that there are some significant regulatory problems regarding training and residency positions, especially since residents have basically zero leverage at all.