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.
Think about all the people involved in health care that aren’t involved in actually providing healthcare. Billing, debt collection, assessing insurance claims, denying claims, legal fighting over denied claims, advertising etc.
All that money is better spent providing healthcare.