So would universal health insurance. The way you do that is to make it an arms-length relationship... in our terms, a crown corporation with a board of directors made up of stakeholders, running on a non-profit basis. All you have to do is run the joint on sound actuarial principles, keep the premiums levied transparent, and it's entirely possible to have things like this run extremely well.
In the old days, health insurance was tied to employment here... well, not quite; instead, paying health insurance was tied to employment, as a line deduction on your pay stub. However, not having a job didn't mean you didn't have health insurance; it just meant that you weren't paying premiums. Also, premiums were not tied to risk factors; instead they were tied to ability to pay, based on your income... if you had more income, you paid more, period. Now, I know that there's a lot of irrationality about that in the U.S. (the fear that someone, somewhere, is getting some kind of social insurance) but history and the experience of the Rest of the West has pretty much proven that you get far far better outcomes that way. Finally, I think that looking at the poll numbers suggests that while your wholly owned government/media complex generate a lot of piss and wind based on a small group of really mouthy people, the majority of people in the US get this... so yeah, I think it would work.