It's not answerable as a simple yes or no, because it depends on exactly what you do.

For example, I would guess that if you redeem BA Avios points on an AA flight and do nothing more, the answer is no.

But the answer might different if you redeem BA Avios points on an AA flight, and then call in to have the frequent flyer number on the ticket changed from your BA number to your AA number. At that point, even though it was booked with BA Avois points, it now has you identified as an AA frequent flyer. (People with AA status often do this to get AA status benefits while flying an AA flight paid for with BA Avios points.)

Remember, it's already been explained that these benefits are linked to your AA account. So it matters a whole bunch whether your AA account is linked to the reservation or not. But there are ways to change what account is linked to a reservation at different points along the process (from booking to actually being on the plane, at several steps along the way). Hence it's more complicated, as in does the reservation have to always have had your AA account linked to it to get, say, a checked bag benefit, or only at the moment when you would (otherwise) need to pay the checked bag fee?