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.
I believe short term there is an increase, give that a lot of people are braving stuff they can’t afford to fix. But it stabilizes and reduces in the long term cause as society becomes healthy, and with other health initiatives like healthy eating, exercise and preemptive healthcare, it normalizes with a general reduction in staff numbers.