I've seen the same thing with my spouse and kids (all no AA status) when booking with BA Avios and then managing the booking via AA.com. The only seats available to assign are the ones in the very back of the cabin, or middle seats. I think it is because the BA Avios redemption comes from the T booking class, which restricts which seats are allowed for free assignment. If all the free seats are all assigned already, then there are none to assign to the reward booking, even though there are other seats available when you make a dummy cash booking.

You can try calling AA Web Support and see if they will assign you a seat, or just check in at 24 hours before departure to select seats.