Yeah I still don't think you're right. The point you're making is understood to the point that it's just noise to add an explainer every time you see the word free.
Just call it universal healthcare, a toddler would think something is free because you do not have to pay for a service immediately. For most people in Europe paying for healthcare is number two or even number one expense, no real concern for the really poor or really rich but kicks the nuts of the lower-middle to middle incomes.
The only people who call it "free" are from the US, not a single european would call it "free".
It's stupidity, not cultural difference, when one side talks about free and the other side who has that "free" health care feels it's a massive (but good) burden.
I'm absolutely for this system, as a Swiss citizen I'm incredibly proud of what we've achieved, but man it's hard...really hard, and it's only going to get worse because of the age shift.