Yeah, I know of no country that has strictly private healthcare that could be called working, even charitably. The closest I'm aware of, though I forget exactly which nation it is, also does stuff like own most of the country's housing; the private healthcare only works to the extent it does due to other parts of life being gov't ran.
Most healthcare, and especially emergency healthcare of any sort, just fundamentally does not work in a private market environment. Can not work. The things that keeps prices manageable and service or product at a decent level (stuff like being able to shop around, having access to multiple competitors, and sufficient knowledge and leverage over providers to negotiate and whatnot, and so on) just flatly is not and cannot be available to a person who is actively bleeding out, or coughing their lungs out, or dealing with chronic pain, or etc., etc., etc. Shit just don't work, so prices will go out of control, service will be shit for most participants (both of which can be deadly when it comes to healthcare), and so on.
Certain other necessities aren't quite as bad on that front, but the same issues arise with most major lifegiving utilities, really. It you're dead from exposure without a roof over your head, whoever controls the housing has you by the proverbial (or literal, depending on how corrupt shit gets) reproductive organs. It's bad enough when a public servant has that kind of leverage on you, someone who's primary incentive is to drain every last monetary unit they can out of you is just... worse.