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.
Plus, healthcare providers spend a lot of time doing financial admin. They also do exams and tests for people that don’t end up having treatment due to cost. Free healthcare cuts down on a lot of waste within the system.
Costs will go down. Even if staffing increases, which likely, if staffing increases, I’d see that as an even better scenario. I certainly don’t see medical staffing going down- it’s the insurance ghouls losing their jobs… and i rather see that as a good thing, too. (we’ll need some admin anyways, so we can keep the not-ghouls.)
Costs can go up or down overall, as more demand is created. It all comes down to what will be covered. Cost per treatment or cost per service should drop significantly.
The USA already spends way more than most other countries for worse outcomes.