Stuff costs money because there isn't enough stuff for everyone to have as much as they want. Breathing air is free because there's plenty of it. Land and water used to be free until things became crowded enough that communities had to make trade offs. Radio broadcasts and a lot of software is free because making additional copies of it has negligible cost and there are people willing to bear the cost of making the first copy.
Health care costs money because there's not enough available for everyone to have as much as they want. The shortage may not be visible or easily definable, but we know it's there because it's not free. Prices are set accordingly. Some people choose not to purchase health care. Shortage solved.
Prices mean some people can't afford to purchase health care even if they want it. This makes people sad. The government seizes money from some people to pay for other people's health care. Access solved.
Shortage created again. Patients respond by consuming as much free health care as they can get away with. Doctors react by charging the government more, spending less time with each patient, or refusing to take on new patients. Government responds with onerous quotas and regulations. Health care rationing being enforced by inefficient and far-removed bureaucrats instead of patients.
Ideally you want consumers to efficiently and equitably ration their own health care. This would require instead of using general taxes to make health care free for the poor, that people's health care prices increase both proportional to their ability to pay and to the amount of health care they consume. If we have enough resources to do cholesterol screenings every year for 95% of the people, then the price Warren Buffet is charged should make him decide against it 5% of the time, and the price some random poor person pays should make him decide against it 5% of the time. Unfortunately, no one is smart enough to make that work.