Who do you think pays for it? TAXPAYERS! It's not free!
And usually in those kind of "free healthcare" systems the private clinics get looked down upon or are even restricted/blocked via laws in order to "preserve the 'free' healthcare system".
Thus the people have even less choice. There are many countries with this kind of a "free healthcare" system.
I'm living in one of them.
Want to go to a specialist? Sorry bro, that's gonna be 6 months of waiting time...
So anyone who is not piss poor will go into a private clinic.