As far as I'm aware, she is correct. Principals have 100% control over the staffing of their buildings. Now with that being said, the smart principals will defer to the FAD to make the recommendation to them; however, the principal still has to make the recommendation to HR, who then does their part, then makes the recommendation to the Board.
FADs are only as powerful as the district lets them be; they honestly have no real power on the campus level. Their job is primarily to support the programs at the central admin level.
I understand every district is different, but this has been my experience in the past, so take that all as you will. I have also been in districts without an FAD, and obviously that's all campus principal.