No, it's a small-ish minority of them. Most are government owned and run.
That said, there's a huge incentive to piss away money holding people so you can justify your budget and use poor conditions to justify increases in budget. And on top of that the contractors that supply government jails are pretty evil too.
So it's really a distinction without a difference at the end of the day, it's all a pretty rotten system.
But yes, there are incentives that do reward cells and even individuals for number of imprisonments. And no, they do not check nor punish "administrative errors".
Taxpayers are paying the private companies, but they're very much for profit.
About that:
https://www.forbes.com/sites/thomasbrewster/2025/01/29/billi...
https://www.motherjones.com/politics/2025/03/private-prison-...
https://afsc.org/newsroom/ice-signs-massive-contract-profit-...
As has been a rising sentiment as of late: "The cruelty is the point"
You're right that it isn't efficient in any sense. But the kinds of people who go into and are chosen for "law enforcement" tend to be the very people that should never be given a weapon. It's just a large scale Stanford Experiment in that regard.
Or maybe in both cases the point is deterrence.
You would think that a country with a whole department devoted to government efficiency could work that out.