This one has a simple historical answer. In the late 1800s, when public education was developed in English speaking countries, it was commonly accepted that society (ie even the rich people) would be better off if poor people were better educated.
I haven’t read as extensively about American plumbing as I have about American Education, but certainly in England and Aus, what I call “sani-care”, that is free sewerage systems for working class people paid out of property taxes , and cumpulsory state controlled sewerage systems for middle-class areas paid by levies and property taxes, was introduced for the same reason.
That is, on the basis that rich people wouldn’t get sick and die young if they weren’t surrounded by poor people with endemic and epidemic disease.
Since improved sanitation made more difference to health than medical services ever have, it is arguable that America already has government controled health care.
Which is another thing that looks so strange looking from Aus. The American health/sanitation/disease control system is in poor health, with problems like drug-resistant TB at 3rd-world levels, yet the rich people don’t seem to be worrying about it very much. From here, that looks like is a very foolish attitude, independent of the medical care debate.
And in several countries, the public healthcare system started before either WW. Initially, a few industrialists realized a healthier workforce was more productive; the difference could be so impressively large that they started being able to convince others; enough of them were involved into politics to make it a country-wide issue rather than a company-wide or factory-wide one, as early as the last quarter of the 19th century.
What’s missing in the US is step 3, but as I say, in several European countries UHC was taking its first shaky steps way before that. It got more universal as funding got better; in some cases, private schemes reached a point where the people who had the right to join them claimed their right to choose not to (many of these have died off, others remain), as they considered the public system better; in others, when someone brought up the option to opt out, it turned out that the majority of the potential members preferred the public system so their scheme got shut down.