If you need dental work, glasses, etc, you better have either cash or a "mutuelle", which is usually tied to your employer (though you can purchase your own above that if you like). Lower-level jobs don't always have one (or it doesn't cover much).
However, procedure prices might be lower than in the US, though (I don't actually know) which is a big part of the issue with access to healthcare.
Here in Japan, everyone has insurance, either private through their company, or public through the government, and the insurance pays a flat 70% of the cost, and the patient pays the other 30%. But the procedure costs are generally not that high to begin with, so the 30% copay ends up being pretty cheap usually.
That's the result of the privatization that has been pushed by the Anglosaxon business lobbies to Europe and everywhere else since Reagan/Thatcher period. They forced privatization of whatever they could get away with. Its still a socialized system with some forced privatization being pushed through. France is not the only country - all US satellites have been pushed to do some degree of privatization.
I think pushing for single payer specifically is one of the biggest mistakes of the American left, because it's that much harder to sell in US due to more restrictions - and that's inherent in the model. If we took something like the German model instead, I think we'd be way ahead by now.
Its socialized healthcare. You dont have to put in quotes. It was first advocated by socialists in the First Socialist International at the end of 19th century. It constitutes part of the social democratic program.