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.
staffing issues aside, generally, “free” healthcare (or rather, government provided healthcare) generally reduces the over all costs- in part, as you noted, by allowing far more prevention and efficiency. Also remember, in places with health insurance like the US, you also have staffing for the insurance agency, and the half-dozen agencies providing support to them, as well.