Yes, this is what I find a little odd about the US proposals. I've lived in UK and Spain and in both they have great free public healthcare for everyone, and if you (or your employer) want to layer private insurance on top you can. Wouldn't that be a much more appealing proposal to Americans?

Perhaps, but given the current shortage of healthcare providers we would probably see many doctors drop out of the public healthcare system and only accept private insurance or cash payments.