Yes, you are very correct. However, when I have cheekily asked for a re-route whilst on AA tickets, the Flight Connections agent at T5 automatically picks up the phone to AA (somewhere).

Once I was told that if I wanted to re-route whilst on a AA ticket, I would have to go from T5 FC over to T3 get the change done there, return to T5 to actually take the flight, all in under 60mins (not even taking account of conformance!). Needless to say, I just booked an Avios ticket from my app, and thanked them politely for their help