It works in a sizable number of countries.

No one is saying food should be free, the government has no business having any control over your diet.

No one is saying auto repairs should be free, the government shouldn't be in the business of deciding what parts go into your car.

Healthcare is different in that:
(I) Public systems are working well elsewhere in the world.
(II) Your current system is problematic and is getting worse. The age at which people begin their work careers keeps getting later, and people are finding themselves unable to get insurance once they are off their parents insurance because at 25 they already have pre-existing conditions. Many of these conditions are environmental (or genetic) and hence outside the control of those afflicted by them. Even if they are managing the condition through proper responsible diet and exercise, its existence makes it impossible for them to get insurance save through getting a job that provides good insurance. These jobs are getting rarer as employers cut costs. Furthermore, with visible medical conditions employers often refuse to hire although they will officially site other reasons.

On the other hand I'm not surprised to see a country/municipality/state that can send firefighters to your house and bill you for it, despite the fact that you didn't want them there, believes in private medicine.

Likewise for a country/municipality/state that sends someone a $28,000 US bill because their house burnt down.