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.
In the UK, hospital administration still goes on. There’s no shortage of jobs for skilled administrators. There are people to assess whether you’re eligible for free treatment, people to assess what treatment you’re eligible for, people to bill patients who choose to go private and chase their debts, etc.
Plus there’s the extra layers of administration at local, regional, and national levels.
Plus there’s people moving money around to ensure bills get paid, facilities are maintained, and staff are fed. Money still moves around even when the patient isn’t the one paying it.