Reroutes are best handled by agents on the ground. I doubt that a BA phone agent would have done this. I also doubt that an AA agent at LHR would have denied this.
Generalizations such as this never work. Just look at the BA thread for the dozens of BA regulars damning BA for its customer service !
As soon as OP told the AA agent that "he was done" that was the end of any of hope of anything beyond a few miles as a customer service gesture.