Causation goes the other way. If your healthcare is cheap, you might as well socialize it, and it will mostly be OKish; if it's too expensive (as in the U.S.) you have no chance of running a satisfactory socialized system. Switzerland has largely private healthcare and mandatory ("universal") health insurance, but its costs are not too high despite it being a comparatively small country with inherently higher costs of providing such services.
> For everything else, it's your health and possibly your life at stake; you can't really say no to treatment, so you pay whatever it takes
This is not really true. There is such a thing as pointless overtreatment, and the right balance of benefits vs. costs can only be reached with increased price transparency on the part of providers.