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.
“Free healthcare” doesn’t exist. You can spread the cost differently. Either you pay what you need - which could be a lot - or you pay less but consistently into a big pool along with other people and then that pool money gets distributed to health care providers. That smaller but regular contribution will go up if everybody goes to see their family doctor unnecessarily so there is a bit of a feedback gauge. It isn’t all milk and honey in socialized health care.
No matter what system your country uses, you will have heard about the same problems. Not enough staff, lacking qualifications, people being overworked and underpaid - in particular on the lower rungs of the ladder. That leads me to think that the staffing levels are about the same. Maybe one system has more work hours invested in preventative care while the other needs more in mop-up crews for those who fall through the cracks.
The indirect savings of people going to the doctor earlier etc actually mean some of the healthcare ends up being free.
Also, saying that you’ll have the same problems regardless the system is both factually incorrect and unnecessarily nihilist.