One thing Europeans often don't consider is that the population and size of a European country is about on par with a bigger US state and that US states have a lot of legal independence. So the variation is a bit less crazy than it sounds.
Similar with healthcare where it's by and large provided by the state.
Obviously speaking from my experience which is limited to the bigger countries in europe
As for our healthcare - I'll keep our system, warts and all. Not saying it can't be improved but at least I know when I need care I will be able to get it in my (literal!) lifetime. And if I'm sufficiently motivated I can get the best care available. I'll take that over any least common denominator one-size-fits-all model any day. Maybe your politicians are just better at running government programs - ours aren't. They are stupid bad at it. All I have to do is look at how well the free government provided health care is for our military veterans and know I don't want our government anywhere near my healthcare, thank you very much.