Secondly, the real limiting factor now is not the AMA but the AAMC[1]. They serve as the current bottleneck for increasing the number of people who receive the MD degree.
However, getting an MD doesn't mean that you can practice medicine. Technically, you haven't even been trained to practice yet - that happens during residency[2]. The bottleneck for the number of practicing physicians (as opposed to the number of MDs) is what is really relevant. That is capped by the number of residency positions available. These are funded through Medicare, so the real limiting factor is the government spending allocated towards Medicare which funds residency positions.
So, blame the AAMC, blame Medicare, blame the government, or blame Congress, but don't blame the AMA, since they aren't the bottleneck for increasing the number of practicing physicians. (And one can easily make the argument that increasing the number of medical school students while keeping the number of residency positions open constant is a waste of money).
As for reducing the costs, that's easier said than done. The amount of time it takes to train someone to (e.g) be a good cardiologist is significant, and the length of time alone makes the costs very significant - trying to shave off a few pennies here and there is really just a drop in the bucket.
The only way to lower these costs appreciably would be to lower the amount of time it takes to train in these specialties, but that comes with a number of risks. Already there have been a number of lawsuits arising from overworked and sleep-deprived residents making mistakes, and it's a tough sell politically to say "Your cardiologist spent 33% less time training to receive a bargain degree. Any loss in quality that impacts your health is just a side-effect of keeping health costs low."[3]
[0] It's also misleading to imply that the AMA seeks to keep income for doctors high. The relationship between doctors and the AMA is complicated, but fairly rocky and sometimes downright acrimonious. Keeping incomes high is (if anything) a side-effect of the AMA's practices, not a goal. (And if it were, the AMA would be failing heavily at this goal, as physician income has been declining steadily for many years now in most states).
[1] https://skeptics.stackexchange.com/questions/4561/does-the-a...
[2] Which only gives you the ability to practice in a limited capacity - any specialties or subspecialties will require a fellowship, which can be as much as 10 years of training after receiving the M.D.