As someone who has flown TATL fairly regularly over the last 10 years, I believe the statement was true before, but not now.
Ten years ago, it was clear that one would pick BA over AA on a TATL flight. It was clear to AA as well because you couldn't earn Aadvantage miles if you flew BA metal. It was also the case that generally F and J on AA metal was substantially cheaper than BA in those pre-NGBC days where there wasn't even a powered seat ajustment in J. Since then, my experience is that I would take AA if it is NGBC or better (or if I can upgrade with miles or a SWU to F). (which is another point, as an AA frequent flyer, I appreciate the ability to upgrade to F). I have flown twice this year in F on the new 3 class plane, and it is a product that is truly superior. (Also the addition of the soup course has been fantastic). The service has also been fantastic.