Supposedly one of the downsides of more socialized systems are that they're bureaucratic, but I think the U.S. system, whatever you want to call it (it's not really either a functioning market or a properly socialized system) is somewhere near Peak Bureaucracy (certainly more than Denmark's, where I now live). It doesn't help that everyone bills separately: if you ever have to visit a hospital, you will get something like 15 different bills, because everyone from the anesthesiologist to the surgeon are apparently independent contractors, or at least bill that way. And you have to go through the same opaque negotiation process with each of them. And many of them are sloppy and contain errors, e.g. billing you for things that insurance actually should have covered.
I'd be fine conceptually with a model where catastrophic care is insured and smaller expenses are paid for out of pocket. But then I'd like: 1) the catastrophic care to actually, 100%, guaranteed be covered, without loopholes, lifetime maxima, excluded conditions, etc.; and 2) the smaller expenses to be priced transparently up front.