The BA ticket wouldn’t necessarily automatically cancel itself, and AA staff wouldn’t have the bandwidth to proactively check if you made the AA flight, so unless you tell them that you made the AA flight, AA staff wouldn’t know to cancel the BA ticket even if AA staff could find out that you made the AA flight.

Also, oddly enough, I have gone standby on an earlier flight and my original ticket for the later flight not only want cancelled, but I also cleared the upgrade list in the later flight... even though I was already on the original flight, halfway through my trip.

So basically AA won’t necessarily know which flight you ended up taking, and even if AA could find out, they probably just have too much to do to monitor your flight status.