But we shouldn't get away from that base level of services that the government provides, and health care, just like the military or law enforcement, is one such service.
No.
Following that logic, there's nothing that the government shouldn't provide. How about food service? Transportation would all be public in your government assigned car. Why not jobs?
There is a theory, to which I subscribe, that the purpose of a government in a capitalistic system is to provide markets that the free market does not efficiently provide. That means, things that cannot be done efficiently for a profit. Among those would be roads, police, military, and fire protection. One could further argue schools, but at a little more of a stretch.
Can you make the argument for health care? I'm not sure. But the current system sure seems inefficient to me. Does that health care doesn't work "for profit"? Not necessarily. But I am willing to consider a not-for-profit alternative.