There's another aspect: In my country, hospitals and (public) health insurance are both operated by the state and work together. If I break my arm, I go to the hospital, show my e-card and that's it. All the financials are directly handled between hospital and the (public) insurance provider. I don't have to worry about cost of treatment because I know it will be fully covered.

Right. The mainstream progressive proposal for comprehensive health care reform in the US is single payer, so-named because it does not nationalize the providers. But the providers are where all the cost is!