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"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."I see no reason why it would work that way.
I mean, switching the US to a completely socialised system is certainly going to be a hell of a task, and even Medicare is way more expensive than it should be, but that's also because it's required by law to be expensive; Medicare is not allowed to negotiate lower prices, which throws away the primary advantage of a socialised system. But from what I understand, Medicare is still generally cheaper than privatised healthcare for the same treatment (though more expensive per patient because it insures only older people).
"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."
The only graph I could find showed Swiss healthcare being even more expensive per capita than Dutch healthcare. This seems to confirm my suspicion that a privatised is generally more expensive.