If you think that jails should not exist at all, then, granted, nobody should be paid to run them. If jails should exist then presumably you can make a list of the things you would like to see in them, and whoever can best realize that list could be justly rewarded for doing it on your behalf.

You're doing economic analysis on something that is essentially political. Jailers don't get paid because they are the best, they get paid because they have government contracts - presumably they made the lowest bid for the job. They simply do the minimum to continue the contract and use their gains to lobby for legislation that will lead to more imprisonment or ways to leverage their position as jailer - like creating markets they can monopolize.

Supposed to be that service providers compete on cost and efficiency and quality, like anything else.

The market can not function if the government is taking decisions (or a limited numbers of companies, but they are limited in their capacity to fail by the money they have- unlike government budgets)

The market works when individual people take decisions: because there are many of them, some take risks, and they all talk to eachother.