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.
You bring up a counter point that I’ve heard often, and my usual response is that most could probably find admin positions in the government’s system, and the rest can figure it out like the rest of us.
The last part doesn’t make for a great sales pitch, so what about those people who don’t have a job anymore?
Keeping people in unproductive jobs isn’t an excuse to keep a drain on the the system and bloating costs of health care.
What do they do in other countries that don’t have those roles? They find jobs elsewhere or train in something else. Of course you wouldn’t be able to reform healthcare overnight, it would take a decade or a generation.