And in countries with public healthcare private hospitals exist too.
The existence of toll roads in a country where 99.9% of roads are free to use has no impact on your ability to get anywhere, just like the existence of private hospitals doesn't change the fact that every citizen can receive free care from public hospitals in countries with public healthcare.
I have another reply to you that engages with all of your points. Feel free to respond to that. As of now, you haven't given any principled reason for which services should be provided free by the government and which should not.
The existence of toll roads in a country where 99.9% of roads are free to use has no impact on your ability to get anywhere, just like the existence of private hospitals doesn't change the fact that every citizen can receive free care from public hospitals in countries with public healthcare.