Our only health care problem is the way we pay for it. The answer is simple: Do what every civilized country has done and nationalize it. Get rid of private insurance; the free market only works when the person paying has choices, and we have none in our system.
The problem is our "leaders" don't have the balls to stand up to the insurance companies, none of whom should even be in business. The conservatives claim to be for "personal responsibility", insurance companies' primary purpose is to negate personal responsibility. Drive like a moron and cause a wreck, the insurance company pays, not you. Operate on your patient while you're drunk, or after being awake for 36 hours and maim or kill him, the insurance company pays, not the doctor.
I would just like to note, that they do pay for it. There is a reason certain demographics have to pay a premium for any insurance: they are high risk. That is part of the business model, otherwise they would be hard-pressed to stay solvent.
Also, not every civilized country has nationalized health care. With nationalized health care comes higher payments for everyone, not necessarily the high-risk users. Even now, with the limited Medicare/Medicaid programs we have, the government doesn't sufficiently pay for the health services it gets from hospitals/private practices. Doctors have lost the incentive to accept Medicare because they don't know if the gov't can follow through with their promises.
Have fun living in your dream world where you see everyone getting free (lol), quality health care, because it isn't going to happen with our current government's policies.