Where did you get the idea it was already owned by private equity?

A public library delivers value to a community. Profit is not necessary for that, and can be argued to actually be harmful to delivering value to a community.

Efficiency might not deliver maximum value either when the community is the focus. Something taking a little more time or involving more human effort can actually be fulfilling for those doing and receiving. Firing the librarian that knows all the childrens' names so you can have a kiosk instead isn't delivering value.