I could see it going either way.
With free access, people would be more inclined to go to the doctor for simple and small things, but in return would probably catch more serious issues early and have better access to treatment, therefor reducing the need for intensive and specialized healthcare.
Without, people avoid going to the doctor for small stuff, but end up having to go in with more complicated issues later on.
The indirect savings of people going to the doctor earlier etc actually mean some of the healthcare ends up being free.
Also, saying that you’ll have the same problems regardless the system is both factually incorrect and unnecessarily nihilist.