Sentencing is decided by a judge not by a prison warden. Beyond vague conspiracy theories I don’t see the connection.
It's not vague at all. Private prisons lobby for mandatory minimums. Mandatory minimums remove sentencing power from the judge: https://en.wikipedia.org/wiki/Mandatory_sentencing
The older system you describe is called discretionary sentencing, it is a dissipating vestige of the era before private prisons.
The prison guards union is an incredibly powerful lobbying force, which has successfully increased prison sentences for many crimes, blocked efforts to decriminalize marijuana, made more mandatory sentences, etc. Whether those guards work in the 92% of prisons which are government run or the 8% which are privately run makes no difference.
I think it would be pretty naive to assume that private actors can have no influence over an unelected/appointed judiciary anyway, but don't forget that many places in the United States have an elected judiciary that needs to campaign every few years to maintain their status, which clearly opens them up to influence by those prisons.
All prisons are required to maintain a certain minimum population to justify their operating budget.
At any rate the entire debate of public versus private prisons is laughably flawed because even public prisons are reliant upon commercial service vendors that cater specifically to the prison industry at ludicrously inflated rates to both inmates and tax payers. And both prison types are known to make use of inmates as cheap commercial labor which undervalues the inmates and robs the local economy of manual labor jobs.