A large subset of the US does have socialized medicine. Medicare (65 million people), Medicaid (75 million people), and the military health system (10 million people).
That’s nearly half the US population.
That’s nearly half the US population.
No private parties at all.
Europe is a big place with many different types of systems. What you're describing sounds like the UK. But places like NL[1] or DE[2] are more like Medicare/aid for all (basic coverage) + Obamacare (mandatory private insurance, sometimes supplementary). The providers may be private practitioners.
NB: I'm not an expert in these systems so I may have gotten some details wrong. My main point is that "European healthcare" isn't a monolith and it's not all state-owned.