Earlier quoted context omitted.
The problem with having non-science people run an organisation of science people is self explanatory. In the Netherlands they have outsourced the organisation and managment of health care to the doctors themselves (roughly speaking). It is often cited as one of the main reasons the Netherlands rank #1 in health care ratings over the years.
In the US, the American Medical Association is run by doctors and they artificially limit the supply new doctors by capping the number of medical schools. So sometimes having the fox guard the henhouse is not the best way to save money.
It's also important to note that the number of medical schools isn't really artificially limited by the LCME, as they will accredit most any medical school that meets the accreditation standards.
On top of that, medical students in the US have to undergo a residency before becoming fully licensed physicians. These residencies are accredited by the Accreditation Council for Graduate Medical Education (ACGME), a separate organization.
However, while the ACGME can accredit these programs, the majority of funding for training resident physicians comes from the federal government (department of health and human services, center for medicare and medicaid services), and is capped by the Balanced Budget Act of 1997 (modified in 1999 to accommodate the rural physician shortage). Today, this costs the federal government around $11 billion each year for all of the residents in the country.
The real question, then, is who we can point to as the bottleneck for new doctors in this country. I found this article enlightening, but it may be behind a paywall: