> What numbers are we talking about?
In 2015 there were 41,000+ applicants for ~30,000 residency positions[0]. I don't have the 2015 numbers for medical schools, but in 2014, there were 18,000 graduates in the US[1].
Increasing the number of medical school graduates in the US would mean that a larger percentage of residency slots are taken by people who went to school in the US but not the total number of residents. Ironically, this would actually be a bad thing cost-wise, since foreign medical schools are usually cheaper and students carry a lower debt load[2].
> Does this mean there is an ever-growing population of "post-doc" physicians, stuck in labs or other non-patient-facing situations?
Yes. You are likely to see them take jobs in industry - e.g. at a pharmaceutical or insurance company, but on the business side, since they're not licensed to practice. (Nor are they even qualified - medical school teaches essential knowledge for doctors, but they receive basically no training on actual clinical practice. All of that happens during residency).
It's worth noting that not all doctors want to practice medicine - for example, people in an MD/PhD program often go into research, and that's a good thing. But if we're talking about increasing the supply of practicing physicians, increasing the number of medical school applicants in the US makes literally zero difference.
[0] http://www.nrmp.org/press-release-2015-residency-match-large...
[1] https://www.aamc.org/download/321532/data/factstable27-2.pdf
[2] These aren't necessarily "foreign" students either; it's quite common for people in the US to go to medical school in the Caribbean and then come back for their residency.