(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.
"There are people not good enough for health care and helping them would violate this natural order".
That doesn’t make it the wrong policy decision. Lots of systems we happily manage with similar dynamics. But I don’t think denying that basic fact is the right path forward. The moral hazard is real and worth acknowledging.
I think bad health is usually its own punishment.