Because US health care (not the system) is the best in the world and the overwhelming percent of people are already covered, I'd first try tweaking the current system system, esp in administration and delivery, before trying to replace the entire system with a new socialized system.
Saying that most are covered is a misuse of the word. I'd go for "conditionally covered" as a more accurate term. A lot people don't realize that their insurance can and will screw them if their illness gets bad enough, sometimes legally, sometimes illegally. (Let's not forget that dental/vision are frequently left out from the start, too)
When the politicians and media use the word "covered" to describe most Americans' health care status it amounts to a kind of Orwellian thought-control by means of language.