Who has the real power?

No, that does not happen.
First of all, bid matching is done via computer algorithm. The sororities submit their ranked lists and the algorithm does the matching between the bid lists and the MRABA cards and generates the final result. A sorority simply cannot come in and alter any of that data to get a pnm. IF that happened it would be computer hacking and very difficult to pull off, not to mention a felony crime in Mississippi.
Second, a pnm being "99% sure" means absolutely nothing. There is no way for her to know her rank on a bid list or to find that out. Not even the general chapter membership has that information; only the recruitment chair and advisor and panhellenic have that info and they don't share it. There is always rumor or speculation or excuses about how or why a pnm went to one house and not another and a lot of times these are made up scenarios to avoid telling the truth to someone's face, on the part of sororities as well as pnm's.
Third, no house gets every single pnm they want. That's part of rush and just how it goes. Good sororities focus on their success and are glad for who they get and move on. No single pnm matters that much.