A few things here, first off, this is not a UM situation, where AA would have been alerted and preapred in this situation. Secondly, being underage, maybe the AA agent trying to help may have had a problem in accomodating her to a room in the hotel, it could have been hotel policy regarding the age factor. 3 hours wait, maybe employee was trying to help out as best as possible, perhaps find a different routing, who knows, also, how could the employee be certain that the other person on the phone is truly related to the minor? They where not given any contact name and number. The last thing you would want is to have someone hand off your child to an unauthorized person.
Finally, cell phones, it seems you where able to communciate with her, maybe via payphone or cell phone, regardless, this could have easily been solved once the MX occurred. A simple call to the hotel to book a room, the clerk will verify with that indeed you are the owner of the card and authorize the person to stay. Seems to me that the main thing is that AA was not able to provide a hotel due to the MX, but maybe it could have been the hotel not able to accept the minor. Whose responsability will this fall under? As neither party may have had a contact name/number for the adult.