The intermediary in healthcare makes a significant difference, as, by going through employers and using insurance, the US market is quite fragmented, and there is minimal alignment pushing prices down. The US healthcare provider doesn't get more business by providing a better cost/benefit ratio: It's easier to splurge, and get business via an expensive, comfortable-ish service.

When one then compares US facilities to foreign ones, it's trivially easy to see that many parts of the system just look different, which comes from the perverse incentives of going through employers that aren't big enough to actually push down on providers' prices at all. Both truly private, low insurance systems and universal healthcare systems end up having much better incentives, and therefore lower prices, regardless of who is paying for them.

We get something similar when you compare US universities to those in Continental Europe. It's clear that over there, the finishing school component is so vestigial as to be practically invisible, whole the focus is a filtering mechanism that attempts to teach something. Go look at, say, Spain's universities and see how many open electives are there, or how many university-wide general requirements exist (0). Each degree is basically an independent unit, and chances are you'll never visit a building from a different school. Undeclared majors? Nope. Significant number of students living on campus? Nope. Sports teams, offering scholarships? Nothing of the sort. This also leads to much lower prices to the school itself, regardless of whether it's all paid by taxes or students.