Which approach is better is very difficult to objectively evaluate. There are a vast number of yardsticks you could use, and I can't see two people ever agreeing on which yardstick is the right one to judge a health care system. Since cancer is one of the top quickly fatal medical conditions, I consider the wait to get treated for cancer to be an excellent yardstick for evaluating any health care system. In this regard, the USA is way better than many countries that have government run free (but rationed) health care systems.
Note I didn't say "better" - I said "worse". For example, there are different voting systems, each that optimize different outcomes. Impossible to agree on the best system because we all think different goals are important. Yet despite this inability to determine the best, First Past the Post is across the board worse than the alternatives. PIck a feature people really want in their voting results, and FPTP is worse at it than any other.
My point being, though, that "every system has its flaws" is a terrible excuse for accepting problems.
Cancer is the second biggest killer in the US. There is no cure for cancer, any cancer. We can treat it if caught early enough by cutting it out. That requires having health insurance in the first place. Almost half the people who get cancer die from it within 5 years and that number is only that good because we're really good at breast cancer. Not sure cancer is the yardstick you want to use as how "great" the US health system is. Even among cancer patients you see disparities in how the wealth survive versus the poor, basically wealthy people do better. We're rationed the same as Europe, we just ration differently and claim it's because of social darwinism.