For what would she be arrested? Trespassing (or attempted trespassing) after the airline rep allowed her to board, or theft of service (or attempted theft of service) after paying for a ticket/change fee/standby?

It's easy to arrest someone, but a crime has to take place if the arrest is going to amount to anything in a court of law. Even with an arrest, some prosecutor would still have to find it worthwhile to take up the matter and not chock up a loss of some sort in court or elsewhere before it amounts to anything. Which LEO wants to be known for making arrests that don't fly with the prosecutors and that flop in court?

The TSA's boarding pass and ID checks at the gate are worthless -- not that they are worth anything at the security checkpoint to get airside either.