The uncomfortable resemblance to slavery and other obviously problematic aspects (https://en.wikipedia.org/wiki/Kids_for_cash_scandal) of it probably has a lot to do with it getting such disproportionate attention. (The vast majority of prisons are still public.) That and it offers an easy, feel-good solution that unfortunately doesn't address systematic problems: just ban private prisons.
even if one is concerned with remarkable similarity of prison labor to slavery, implications of 13th amendment clause, etc., most of the prison labor is for wholly-government-owned enterprises, frequently manufacturing things for other government departments.
As another example, Louisiana is phasing out private prisons. That should be great! Meanwhile Angola (state-run) continues to have prisoners picking cotton.
I know people know about this because they always bring up prisoners picking cotton in these conversations, but then the talking point remains "private prisons" somehow. So I still don't get why this idea is so sticky.
