My understanding is that the company you booked the flight with (here, AA) bears the responsibility of notifying you of any cancellations or changes in the itinerary. They are acting as your agent for the booking.
Even though the flight was canceled by BA, the change in your itinerary (assignment to a different flight) was made by AA. BA cannot “touch” the booking made through AA, as the experts on FlyerTalk put it. Thus it was AA, not BA, whom you contacted to get a flight you preferred over the one that was assigned to you, correct?
The fact that you paid BA for the seat you wanted on the original flight, using the BA booking code to access Manage My Booking, has nothing to do with it. That is not a change in the itinerary, just a perk that comes with the booking. You could have ordered a special meal that way too, if you wanted. But you could not make changes to the booking itself (like changing to a different flight) in MMB.
We took our daughters to Italy some years ago for a graduation trip, flying business class on British Airways as a treat. I did not have enough BA miles for all 4 of us, so I booked 2 seats with BA miles and 2 seats with Alaska Airlines miles. Not my preferred way of flying, because of the separate tickets, but our only option to get us all in business class.
We flew into and out of Zurich, with a layover in Heathrow in both directions. About a month before the trip, BA changed the time on the Zurich to Heathrow leg for the return trip. I got notification of the change from BA for the BA-booked seats, and from Alaska for the Alaska-booked seats. And because we were on separate tickets, we ended up on different flights, with the flight assigned to the 2 of us booked with BA miles being preferred. I had to call Alaska to get our daughters’ flight changed to the same one we had. Fortunately there were still award seats left on that flight.
Another example—-we use our Alaska miles to fly all over the US, and often the itinerary includes one or more segments on their partner American, who serves some of the smaller regional airports that Alaska does not. When AA changes the flight time—-which in my experience they tend to do a lot—-we always get notification from Alaska, never from AA.
I am going into some detail here because people need to understand that when they book a flight with Airline X that includes a segment on Airline Y, ( a partner), it is still Airline X that controls the booking and the responsibility for overseeing it. It is not the same as booking directly with Airline Y.