The government delivers citizens into the hands of private management all the time: required insurance policies, training programs, licensure, etc. Mandating that every obligatory service be not only funded, but implemented, by government would extend its rolls of employees, budgets, laws and legal complexity, competencies, and requisite voter knowledge beyond any reasonable possibility.

You seem to be ascribing a negative to the mere profit focus of a private company, rather than the outcome. We could do the same for insurance, medical practice, automobiles, plane travel, lodging, and everything else important; but we don't, because private companies in the open market perform better.

Would you then deliver everything important into the hands of government monopoly, despite everything we know about the benefits of competition, for the moral premise rather than results? And if you think the results weigh against private prison companies, might we restructure the government/private arrangement before nuking wholesale the notion of the market?

WRT the incarceration rate, I'd have to see something linking that to private companies running prisons, rather than problems with the justice system itself.

Consider the abysmal performance of government-run schools, among others, as a counterpoint.