The problem is residency, not the medical schools. The medical schools actually turn out slightly more graduates than find places in residency programs. And without completing a residency, you aren't allowed to practice medicine.
Proposals to expand residency run into the problem that it costs the government money. Residencies are funded at the federal level; teaching hospitals get a certain amount of money per resident to compensate for the effort required to train them. And the number of residencies funded per year has (I think?) been maintained at a pretty constant level compared for a long time, with so many new doctors per year per million people.
One option, I suppose, would be to allow self-funded residencies. Why should only the government's money be good enough? But that would probably be an option only for wealthy students, or those who are willing to go even further into debt.
Another option would be to replace residencies with something more like conventional apprenticeship. Is there any reason a new doctor has to learn the practical details of her trade at a hospital, rather than working under the supervision of an experienced doctor in private practice?
Of course, nothing like this is likely to happen. The whole medical industry is extensively regulated; those regulatory barriers are wide moats that ensure fat incomes and profits, which in turn create strong interest in keeping things as they are. Barring utter crisis, nothing is going to change.