You lie to yourself when you write: “Every country, though, has figured out some way to pay medical service providers more than automobile mechanics so that the poor get some access to the care only the rich can afford and to minimize public health risks affecting all classes.”
The sad truth is that everywhere in the world the rich have better health care than the poor. People there just accept it, and do not talk about it as much as we do. Also, those other countries have codified lists of services available. People there accept that even if there is a cure for a given illness, but if it is not on the list, it is force majeure. Most often, they simply do not know that the better option is available.
Every concept of socialized healthcare, as you propose, leads to lowering quality of service with increased cost. I recently reached a Medicare age. In every step, I see increased costs, and doctors stimulated to prescribe procedures that Medicare approves, and steering me away from procedures not approved. In my personal case, I found out at least one instance of a very economic and effective medical procedure, commonly used in Europe but not approved in the United States. My humble suspicion is that the reason for this is the fact that European manufactures of the equipment to perform this procedure are much more advanced that the American ones. My guess is that when American manufactures will catch up, Medicare will approve this procedure. In the meantime, American doctors do not even mention to patients about this possibility.
I can agree with you that we should have a system giving the poor and disadvantaged minimum quality health care. But we should not lie to ourselves that it should be the same care as riches can pay handsomely for it. It should be the basic medical service, but so rudimental, that those who can afford to pay for a better service would not be willing to accept it. The point is in having a better service that most of us can afford. But this can be accomplished only in the free market environment.