The Brain Drain involved scientists and engineers who saw defense cutbacks in the UK impacting their lives. They were welcomed to the US and many wound up in NASA. But they were English coming to an English speaking country and so their cultural adjustment was lessened. They were also quite sought after by industry and the DoD.
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.
I checked it out. Reagan was right. There is nowhere else to run to. Switzerland seemed the best choice for a doctor though. Everybody has to buy insurance but policies are private market, like car insurance here. Great if you're a doctor but sucks if you believe in freedom.
(Costs keep going up since you can't opt out and more requirements for coverage keep being added)