We see how that works. Governments take 40% or more of your income, or insurance companies have outrageous rates. And then the bureaucrats keep part of your money for themselves, and then tell you whether you get healthcare or not by restricting use to the money you paid them. Why should I pay for government administrators to have huge salaries just to manage the money I gave them?
I would rather pay for a doctor's visit out of pocket with cash, and earn interest or invest the money that would otherwise go towards taxes or insurance fees in my own health savings account.
Most people can afford routine healthcare costs. It is the major medical that is the problem. For that, you need something like insurance, either run by the government, a cooperative, or a private company. You also pay people enough where they can put money away into a health saving account and retirement account. So wages would need to rise for that. And you need a safety net for people who cannot afford insurance, funded by taxpayers and charities. You don't have to centralize things to do any of that.
Routine healthcare should never be free, except for those in poverty. It just leads to inflation and rationing of routine healthcare.
25 years ago, health care used to be more affordable. That was before the insurance companies paid for everything, and before hospitals were privatized and consolidated. Doctors could not charge a lot because people paid cash, and people would choose doctors with affordable rates. But now with copays, people think that the doctor costs $30. So the doctors charge hundreds of dollars to the insurance companies, knowing that if the patients paid cash themselves, they would refuse to pay such high charges. These higher fees just get passed back to the patient in the form of higher insurance premiums. So insurance is inflationary. The cost of healthcare is pushed up.
Government insurance tries to handle this by putting caps on what providers can charge, but you still have the problem of rationing of healthcare based on available funds, and it also gives bureaucrats control over your healthcare. It has all of the same problems as private health insurance, except it is run by the government. And if it is centralized, you can't go anywhere else for a second opinion. If they say no, you are screwed. So that is not ideal either.
If you abolish all private healthcare, then you only have government clinics, and the problem with that is that they can deny you care if they don't like you (a political dissident) or if they don't have the budget to pay for everyone's care.
Instead of any of those, you need some kind of system that is not inflationary, is affordable, and that gives people choices in their care. If one provider says no, they can go to another. The current system is really bad, but most of the alternatives that people suggest are just as bad or worse. If you want a better system, it must include patient choice.