Yes, they would...but lawyers oppose the first and doctors the second. It's a tough sell.
And it's not enough.
Modern medical care if fundamentally very expensive. There are not enough dollars around to make it available to everyone without limits. That means hard choices have to be made.
I would begin by identifying those services which society has an overwhelming interest in providing to everyone. Innoculations against communicable disease, child care for the first year or two of life, etc. These should be provided free to everyone.
I would then look at those areas which gobble up a disproportionate amount of tax dollars. More or less unlimited care for elderly old vegetables comes to mind. If a person hasn't managed to provide for his old age after 40 years of work it shouldn't be society's problem.
What about people who refuse to take care of themselves? Alcoholics, smokers, drug addicts, promiscuous homosexuals, obese junk food addicts. Let them, too, take care of themselves.
Deal with the above - and a few other such categories - and everyone could afford health insurance.
Ah - liberalism. The culture of death.