Earlier quoted context omitted.
Fewer than 10% of prisoners are held in private prisons, so how are private prisons the 'central problem' with the US prison system?
You misunderstood my point. Both government-run and privately owned prisons are paid (or funded) based on how many inmates they have in custody. I'm suggesting that private prisons could be solution to the problem if we change their incentive structure.
The government gains an incentive not to jail people unnecessarily (loss of revenue). Prisons get an incentive to rehabilitate and educate.