An interesting choice of example. The USA is just about the only otherwise-civilized state that largely leaves healthcare in private hands, and "only a few are aware" (as you put it) that the US government spends about as much per capita on healthcare as, say, the UK government -- to be more precise: only two other OECD countries have more government spending on healthcare than the US -- and that total US healthcare expenditure is hugely more than that of any other country, and yet its actual outcomes are pretty much on a par with all those other countries that are spending so much less. (E.g., if you order countries by life expectancy at birth, the US comes in at about #50.)

Of course this is a complicated business, obviously affected by other things besides the public/private difference. But, to say the least, the available evidence doesn't offer much support for the idea that a welfare state providing health insurance ends up being more expensive and less efficient. The only good example we've got of taking a different approach goes quite the other way.