For a little more complete reply:
There is NO compelling case for government intervention in health care for ANY reason, least of all for "efficiency". A very persuasive case can be made that previous excursions by government into health care is the predominant reason for high health care costs in the first case.
Nor can there be a case made out for government intervention in the interest of "moral justice" whatever that might be. Government intervention will always and inevitably result in lowered standards, reduced access, higher costs, and more sickness, disability and death, across all income levels and social strata. How is this moral? Or just? It's just nuts.
Finally an "egalitarian view of health"? What the Sam Hill is that supposed to mean? That we all get sick equally? That we all suffer the same level of pain in our joints?
Illness and health is in fact the most UN-egalitarian of things. Illness mocks egalitarianism in that it is random and implacable. It may strike both rich and poor, but those it strikes cannot take comfort in that poor fact.
The fact is that Marxism is dead except on college campuses and in the fevered brains of some so-called economists, most of whom can't hold a true economic thought in their heads long enough to write it down.
I can think of little besides defense that government does, which cannot likely be done more efficiently by the private sector. The recent privately funded spacecraft is one example... private enterprise got a ship designed from gounrd up and into space for less than the cost of rolling a shuttle to the pad.