Unfortunately it's impossible to make a proper comparison since essentially all developed nations except for the USA have adopted some sort of socialized medicine or (i.e. Switzerland) an extremely regulated private system. So USA has essentially been alone in its experiment towards private healthcare.
Given that the one system which is much more privatized than the rest has had such poor efficency outcomes I would say that the burden of proof is on anyone who would claim that privatizing even more wouldn't just make the problem worse.
