I don't think state funded healthcare works the way you think it does. The american system is the most economically inefficient system out there, to the extent that people without experience of other systems likely end up with highly distorted perception.
Note that a mixed economy (combined public/private funding, like the french and australian systems) are probably for the most part the most economically efficient. A big problem in australia is over-provision of services, especially ending up getting more pathology tests than strictly necessary.
Note that a mixed economy (combined public/private funding, like the french and australian systems) are probably for the most part the most economically efficient. A big problem in australia is over-provision of services, especially ending up getting more pathology tests than strictly necessary.