You started well and then you went on to fall into the trap that so many fall(to think that "just this once" we should trust government to do something for the "greater good"(whatever the hell that means...)).
But again, the greed that you mention is good... The problem is "Regulatory Capture"(which is the situation where you have governments joining forces with private companies and persons, this is the source of problems).
The systems you quoted are shit.. I've lived in Europe and I can tell you that even though it might be good compared to some countries(third world shitholes), it doesn't work and it is a time bomb because the system cannot sustain itself because of it's costs and inefficiencies.
You want to lower costs and make better services? Do away with every regulatory agency(such as the FDA, among others) and cut the political power of unions(such as the American Medical Association), let doctors from all over the world come and compete between themselves in the US, end bureaucracy to open up clinics, hospitals, health insurance companies + 0% corporate taxes and see the companies and health care professionals(driven by their greed) compete with each other for the clients/patients... You'll see what this does to the costs and quality of health services(I'll give you a hint: think Uber vs Cabs, it's exactly the same thing

)...
I'm telling you this because you really started well and did not strike me as someone who is just like so many morons out there that believe in fairy tails told by communist idiots...

You don't have to believe me, just use logic and see that, just as you suspected, everytime you give power to governments, what you get is higher prices and bad services... And the problems you have now are a direct consequence of this: too much government(FDA, AMA... as mentioned).
When what you suggested is done, the people that always suffer the most are exactly the ones you are trying to help.