Divide health expenditures into three categories:
1. "public good" health care (mostly prevention of infectious disease, provision of clean water, and so forth)
2. care whose benefit to the patient is very large compared to its cost (fixing broken legs, antibiotics, or any potentially curative treatment of children and working-age people).
3. care whose benefits to the patient are not large compared to its cost (fill in your favorite draconian examples here).
Categories 1 and 2 can in fact be taken care of by publicly subsidized care at modest cost -- if you don't want to believe me, read Amartya Sen's piece from 1993 in Scientific American on Kerala (India) vs. Harlem.
But on the whole single payer schemes for doing this simply cost too much in terms of foregone medical progress due to the difficulties of central planning.
In the US, public health (category 1 expenditures) is mainly a state and municipal responsibility, and they do it well enough.
The US is rich enough that it can afford to subsidize category 2 health care, and if governments concentrate on that more thoroughly they will do the job better and at less cost. But it would be best if actual provision was handled by competing HMO's, each offering a modest basket of high benefit to cost services, with membership in these funds subsidized for low earners out of taxes, and mostly out of state taxes. People who wanted fancier surroundings or more rapid access to specialists than the basket requires would pay out of pocket or buy supplemental health insurance coverage. This would be an American version of the Israeli system, and that, too, works well enough.
But no scheme can pay for all the care that has a reasonable prospect of providing health benefit to the patient. And it is most reasonable (and equitable) if paying for care that neither (a) improves the health of the public (as opposed to the patient), nor (b) provides a benefit much larger than its cost, is the responsibility of the patient and his or her family.