I just checked my usual carrier, WN, and this is their T&C:
Children ages five through 11 traveling without an accompanying Passenger age 12 or older must travel as Unaccompanied Minors (UMs) on Southwest Airlines.
If the other airlines are similar, at thirteen she could have been unaccompanied and a minor, but not within the technical definition of OM by the airline.
Complete WN T&C here.
The TSA actions are no less reprehensible.