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.
There are other disincentives besides price that keep people from going to the doctor if they don’t need to
I used the health care services at my home town once this summer, and I spent two months there. Basic care is free at the point of service there. And I only went because I had had a concussion. Didn’t seem that the other people were there for no reason either.