Is it really wasteful? I think that is just your moral intuition. How much of current US health care is run by the government via Medicare, Medicaid, CHIP, Veterans Department, and so on? Is these things inherently more wasteful than private insurance? Furthermore doesn't private insurance have a self selection phenomenon where they can choose the healthiest patients in the private market, or they do the employer market which is an larger risk pool. When you get to pick your customers it is easier to have a non efficient market when it comes to risk. Yes the 2010 Health Care Reform Bill made it harder to pick your customers but it still happens.

Furthermore why limit ourselves to the US perspective, there is private and public insurance of several dozen different countries outside of the United States?