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.
Also, here in Connecticut (the place where the rich historically sent their kids to school), there are plenty of public schools that are at least as good or better than at least some nearby private schools. When you get into preschool, for instance, the public schools are a HUGE improvement over all but the top-tier private ones. Likewise, the "magnet" program schools are a public option that is as good or better than most private schools nearby (they are special, well-funded by the state, and use a lottery to get in, though).
This is coming from someone who agrees with you in principle, too. Reality is nuanced, though.