Agreed, even though it doesn't affect FF in AA's case, this is still a very important decision that most people overlook. Another important consideration is which airline you will have credit with when cancelling a nonrefundable ticket.
When booking direct, you'll get a ticket from that carrier.
Other times, there can be the option to choose. If you don't choose, you will get the default: seeATPCO logic for determining the default validating carrier (page 10).
TAs may easily make a manual selection, per IATA Resolution 852. Generally if the fare rules allow a carrier and at least one segment is marketed by that carrier, that carrier can be selected (there are exceptions). Because of the TATL joint venture, choosing between AA and BA is usually easy, even if both TATL segments are marketed by the other carrier. Unfortunately TAs often silently choose based on their own preferences (e.g. commission), not what would work out best for the passenger.