Government Run Hospitals A Solution?

What if we just created free government hospitals that would treat any US citizen? We could have doctors work there with their med school tuition fully paid for by the government, with the stipulation that they have to sign a contract to work in these hospitals for a certain amount of years at a lower salary. After their contract ends, they could go to a private company if they please, or they could stay in the public hospitals. Furthermore, in these hospitals, doctors could not be sued for malpractice and only generic prescriptions would be available. There would be a tax for everyone, but those who had private could write off a majority of it.

To me this would do a bunch of things:

-It would provide access to all citizens.
-It would allow private hospitals and insurance to continue existence.
-It would bring competition in prices to private companies/insurance.
-It would reward the best doctors who get offered jobs in private hospitals.

I know it's kind of different, but I feel like we need to think outside the box. Universal healthcare will not work in America, but I'm afraid we are heading there fast if we cannot come up with something that works for the majority of Americans.