On the other hand, price controls work remarkably well in medicine - look at medical costs in the European economies.
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.
Your comment is correct except this sentence. In reality, if the US implemented price controls, it would simply reduce the negotiating power of European countries. Prices in the US would go down, and prices elsewhere would increase. The pharmaceutical profits would decrease on net, but there would still be plenty of incentive for R&D (much of which is already derisked by the massive amounts of public money spent on it with zero expectation of return).
I don't know why people rail against Pharma when healthcare like hospitals have huge returns on investment in comparison.
https://www.morningstar.com/etfs/arcx/xph/performance
look how much pharma index trails healthcare
Also: generic shortages. Generics are a low-margin business with maybe 6 suppliers workdwide. If one facility goes off-line for this or that reason there's going to be a shortage, and slack capacity is punished by the market. Right now the cisplatin shortage is in the news, and again the market is no help.
That's called a monopsony: https://www.investopedia.com/terms/m/monopsony.asp
"A single buyer dominates a monopsonized market while an individual seller controls a monopolized market."
That's true in some markets, but not all.