I lived under the French system for years, and it’s entirely untrue that there is no cost or choice component. You pick the doctors and you pay them fully out of pocket. Your insurance company reimburses you at ~80% of reasonable and customary rates. If you wind up going for inpatient care it’s different, as is the situation for those in extreme poverty. Also, you can pay extra for 100% coverage if you want, but ISTR it was expensive and not really worth it for us.
Now, when I left, the doctors and insurance companies were setting up an analog of the US system: If you saw certain doctors, they would not charge you at point of delivery and there was no co-insurance requirement. It was still new and inconvenient, so I don’t know how that’s worked out.
I don’t have direct experience with the Swiss system, but from everything I’ve read, conversations, etc., it’s a slightly cheaper version of the US system. You buy leveled insurance, you have to pay some out of pocket, etc.