In most other countries, the prison system's job is to reform the prisoners.
In the US, it is designed to be punitive, and there is a direct financial incentive to increase the percentage of prisoners that are sent back to prison after serving their time.
The system is working as designed: the US has a much higher percentage of its population in prison than most other countries, and extremely high recividism rates.
You're really overplaying the financial angle. Only 8% of prisons are private. And even then, it's the government that has to pay the prisons. Imprisonment always costs tax money.
In the US, it is designed to be punitive, and there is a direct financial incentive to increase the percentage of prisoners that are sent back to prison after serving their time.
The system is working as designed: the US has a much higher percentage of its population in prison than most other countries, and extremely high recividism rates.