The health systems in Europe are not radically different from the US system. The efficiencies of each are difficult to quantify without context. For example, the US invents most of the drugs and medical technology used by the world. Would this still happen if there was more invasive regulation? We can't know.
Besides all of this, think of every other area of the market where the government insinuates itself. Are public schools better than private schools? Almost never. Is the US postal system better than FedEx? Of course not. The government is not a commercial entity. The incentives and influences on it are not conducive to producing quality products at good prices.
People state this as if it's some obvious truth, and it makes no sense to me. My statement would be:
Government is a mechanism which we, as a group, have decided should handle certain functions of society.
It's the logical extension of a group of people in the jungle deciding "you know what, it would be awesome if we got together and built a trail between all our huts so we can visit each other."
Part of that agreement might be "hey, when these barbarians show up we should all protect each other", but to arbitrarily say that the only purpose of government is those kinds of agreements I think speaks to a very warped sense of community.