No, I literally do not know what went wrong or when. I'm not trying to make a point about government-provided anything. Although government-provided healthcare began quite a bit earlier than that, as early as the 1880s in Germany, but definitely in the 1940s when the UK NHS started.
Your final sentence is just the same point upthread: you want a simple coincidence[2] to make your argument.
[1] There is a ton of analysis and argument to do there too, but suffice it to say that "because medicare" is just as terrible an explanation.
[2] It's not even a coincidence! Desire for government-guaranteed health care came about precisely because of and simultaneous to the availability of late-life care that actually worked. No one wanted medicare in the 20's because there wasn't anything to pay for even if you had it. But they damn sure wanted their walkers and hip replacements in the 60's.