(1) It’s expensive (2) Everybody has to pay (3) The government’s gotta run it
But there are plenty of countries with functioning healthcare systems that are private? The Swiss, for instance. Moreover depending on what counts as "government’s gotta run it" (paying for it? administering it? actually providing care?) you can argue that the German or even Canadian systems aren't government run, at least to some degree.
The difference now is the republicans have changed, and nuanced issues are just not welcome on the platform of a party following a cult of personality.