It’s not as if healthcare costs have some inherent reason to increase along with wealth.
Well, there are several big reasons. For example, doctors and nurses in the USA have much higher pay than those in Europe. Part of this is because of policy differences (e.g., the supply of US doctors is artificially restricted by the AMA). But part of it is simply that educated professionals are paid much more in the USA than in Europe, and it's nothing specific to healthcare.
The point is that when making comparisons between US and other rich countries, the first thing you have to do is to account for the fact that Americans (i) have higher GDP per capita, and (ii) have higher levels of consumption even after compensating for GDP per capita. That should be the first-order effect, with stuff like the public-versus-private issue as second-order effects.
I do agree with the benefits to the US not being proportional to the cost, to a point. Lots of healthcare spending goes into things that don't really benefit aggregate outcomes, like heroic interventions that end up extending end-of-life by a few months, or treatments that only benefit one-in-a-million conditions. But this is not just an issue for the US; for example, the UK spends 18x on healthcare per capita compared to Thailand, for 2 extra years of life expectancy. And those individuals who get their lives extended by a small amount, or get their rare condition treated, may have different views on the matter.