We may have fundamentally opposing ideological ideas here, but I see what you're asking. I would actually say that the job of Society is the betterment of the state. Ultimately it is people in society that create and man their bureaucracies; the government is infused with the skills and morals of the people that form it, and will not exceed those abilities.
I don't think that you could ever form a wing of a government that was capable of rehabilitating people without that wing being fully made up of very intelligent people with good intentions; and if you were able to assemble that group of people, they'd do a better job as a private endeavor (maybe not for profit, perhaps as a non-profit that only pays their salaries, etc.) I do think you could make a form of a government that was capable of keeping prisons far safer than they are now, as that's a much more discrete and amoral task that ensuring that convicted citizens (often low-IQ, often sociopathic, often abused, often with PTSD) are rehabilitated.
If you decide that a condition for access from outsiders is to rehabilitate people (with whatever metrics you can come up) then you have just privatised the job of rehabilitation... For what gain? Society as a whole would benefit if prisoners are rehabilitated and find that criminality is not worth it, isn't the job of the State the betterment of society? Why should we create convoluted ways to privatise that?