All other things being equal, maybe. But:

1) A prison is far more easily sold to the population that hospitalization or cures ("what, you mean we're paying for those criminals welfare?!").

2) Imprisonment is "business as usual" -- so less friction and investment needed to move things there (they already are like that).

3) Lots of prisons also exploit inmates' labor.

4) Prison keeps the annoying (to those people) black minorities at bay.

Drug development is extremely risky. A privately run prison is pretty much a government money spigot.

Presumably there is little overlap between individuals or organizations that can provide mental care and those that can provide prisons.