Can a family member who doesn't have POA discharge a family member from the facility permanently?

Your mother can sign herself out of AL. Does your mother need in home caregiving? who is going to provide that? Have your brother read the threads on this site about folks who try the "do it yourself" way, especially if you Mom has dementia.

I would guess his plan is to have her sign herself out. She can do this unless she has been declared incompetent by a Judge.

If you Mom has dementia and she is begging to go "home", you should all just know that "home" is not the place she lived most recently. It means that she wants things to be "okay" again, like when she was young and had parents to take care of her and siblings to play with. It's very sad and we hear lots of stories on this board from folks who thought they were doing the right thing. "Home" is a state of mind and not a place. Most dementia patients I've known who were being cared for in their own homes constantly demanded to go "home".