It's already a bunch of private companies.
> These private companies would drive up the cost even more.
Other way around - by having to actually directly compete for customers, instead of just having to convince a few large corporation prices would go down, not up.
Although we really should not ignore that insurance companies are not the drivers of higher costs, it's health care providers that do that.
It's enjoyable to blame insurance companies, but the reality is their profits are capped by law - they are not the problem. Dr.'s will have to take a pay cut, and there will have to be mass layoffs, there's no other way to reduce costs.