Since this is a comparison between entire economies --- not just health care but the impact on economic mobility as a whole --- an inconvenient fact for this analysis is that PMC Americans are generally better off under our private insurance system than they would be if they relocated to the UK; as in, they come out financially ahead, after pricing in insurance and provider fees.
Another problem with this analysis is that the UK's national health system isn't the only example of a major developed country with a functioning health system. Many countries without the US's problems have private providers and private health insurance.
It's plainly obvious that the US has severe problems. Ideological polarization has fixated the discussion on a contest between public single payer and everybody-for-themselves private insurance. But the core problems have not much at all to do with insurance. They are, probably:
(1) Most Americans get their health insurance through their job, as a path-dependent consequence of bad 1950s tax policy. This in turn insulates insurers from their customers and breaks most of the mechanisms an ordinary market (like food, or transportation, or tax preparation) has from going off the rails.
(2) Americans pay doctors (and providers generally) anywhere from 2-5x more than other countries do. Providers account for the overwhelming majority of all health spending; compared to providers, drugs are almost a rounding error. This in turn is probably a consequence of policy decisions we made in the early 1990s, when the AMA lobbied Medicare to cap the number of residencies funded every year, which in turn generates artificial scarcity.