If she's hogging all the funds to advanced sciences, she probably wouldn't really make a good director though. 
I just get the sense that she's a bit despondent and slightly depressed in Fallout 4. Possibly because of what happened with James in the CW, and possibly because she's not sure about what happened with the FEV lab incident. Its not really something that's said aloud in-game, I just get the sense she's disinterested in leadership. And would prefer to be left alone.
I think Allie would have been named Father's successor without the SS coming along. Father seems to trust her the most, and he basically tells the SS she's someone to rely on on his deathbed. That she's always been loyal to him and The Institute.
Although come to think of it, Holdren seems to be the only one who knows about Father's terminal cancer. Though that may just be because he's Bioscience head (and thus de-facto head of medicine) and not because Father necessarily trusts him.