Are you familiar with the term Brain Drain? It was invented by the British after professionals packed up their bags and left the country. That's what some Doctors will do. They will go overseas.
American doctors could go to the UK or Canada, but they would find the same conditions they were trying to escape. They may not be well received by their colleagues. (many Brain Drainers weren't either). Where else where a US MD could uproot his family to? India, Singapore, Mexico? If the UK had a private health system I'd accept your argument.