I compare public responsibility for health to public responsibility for fire control.
Fires can wreak havoc upon many beyond the fire's point of origin. It's also unprofitable to run a fire department, so given the public benefit, the public foots the bill for keeping a fire department.
Health care isn't the same in that it can be a very profitable business, but because profits are tied to consumption of goods (e.g., medicine) and services (e.g., operations), the industry has incentives to promote overconsumption of both. Meanwhile, the health insurance industry has incentives to deny coverage as much as possible in the name of profit. You might think this would cause the health care industry to operate efficiently, but health care is complex and not easily overseen, and even if the health care industry ran efficiently, the health insurance industry would still have incentives to deny coverage past the efficient rate.
But the effects of letting people who genuinely need treatment go untreated are nevertheless much like fires. Not only does it stand an excellent chance of reducing their productivity and overall employment prospects, the consequences it can have for a family can likewise be disastrous. That doesn't do them any good and it certainly doesn't do our economy any good. So while the health industries might be quite profitable, the net benefit to society is considerably less than if the profit incentives were properly aligned to maximize health instead of consumption and lack of coverage.
But, you know, some folks just have to be sacrificed on the altar of the completely free market, in the name of their own good.