The US government (that incompetent, evil thing you hate) puts a lot of money into research, so in some areas we do have some advanced treatments that people from other countries may come here for, but that has NOTHING whatsoever to do with our private health insurance system. Other countries also have treatments that aren't available here, and in those countries, the citizens don't have to be rich to get them.
I live in the US, where, because of the "best health care in the world", I've lost over fifteen years of my life, homebound because of a serious, life-threatening illness, which is controllable with the right medications, but for which I can not afford treatment. I could have had a family and a career, but because of "the best health care in the world", I've lost all hope of ever having any kind of productive life---and there are thousands of others like me.
In addition, I've also been told I'll probably die due to some serious dental issues that could have been taken care of years ago, when they were minor, if I lived in Cuba, Venezuela, or just about any other country outside of the third world, but because dentistry is even harder to get here than medical care if you're poor, people like me die for lack of care.
So much for your idea of "the best health care in the world".