The point is that Europeans seem to believe that the US does not have a public healthcare system, it does.

I am not sure what your point is about covering fewer people either. The point of public healthcare systems is that there are redistributive, correct? The reason the US public healthcare system does not cover everyone is because there are people who can pay for their own healthcare...which is the same in Europe. I live in Europe, in a system with "free healthcare", I pay $100/month for private healthcare because queues for most things are multiple years long AND I pay $1-1.5k/month for other people to use the public healthcare system I can't use.