What makes price controls on medicine work is the other side of the equation: Massive subsidies to supply. How expensive is it to become a doctor in Europe? How many are trained every year? Who builds the hospitals? Ultimately it's not a matter of price controls, but turning a large percentage of the whole thing into a public service. And even then, it's not hard to find some issues, it just works better, in most ways, that the US' bad mix of public and private incentives and regulation.

When we discuss price controls, we tend to be talking about situations where the supply of goods is provided by the private sector. Then price controls often lead to supply shortfalls and black markets. It's easy to understand if we make labor to be the good with a price ceiling. Many american companies would love programmers at $20/hr, but they can't find any. Imagine that the government caps said salaries at $20: We'd see fewer people going into the field vs something easier, or that just was allowed to pay more. The companies that still get $200+ worth of value for programmers would still want to pay more, but without the supply, they'd try to skirt regulations by becoming more competitive in indirect ways. Maybe your benefit package would include a mansion, and an expensive company car, and a live-in staff. Keeping prices down when demand vastly exceeds supply is very hard.

There is such thing as excessively high prices though, via monopolies and regulations that force waste. I think that's a bigger reason for the US' healthcare pricing problems than the magic of government healthcare. What socialized medicine does is make sure that even the poorest can afford it, which can be seen as a valid objective onto itself.