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.
Until recently. This means that all people can obtain health care regardless of their ability to pay, and that the system is flexible enough to continue to experiment with the ideal implementation.
Are public schools better than private schools?
Public schools are exactly 100% better than private schools for students who can't afford to attend private schools.
Is the US postal system better than FedEx?
They are if you want to get a letter delivered anywhere in the world cheaply.
You have made a lot of good arguments for socializing care here.
2. Your argument implies that there wouldn't be low cost schools in the absence of public schools. The evidence is overwhelming that the opposite is true.
3. I live in Panama. When I need something mailed here I use FedEx or DHL. The USPS is unreliable and takes far too long. If there weren't laws against private first class mail in the US they'd be undercut in that market as well.