The simple solution is that if AA has no seat that they are willing to assign for free, or if no seats at all are available, then you should be able to do an online check in and given a "priority verification" card, the same way that a ticket counter agent would.

A few years ago, my Mom, GLD at the time, was on a delayed inbound flight on another airline, connecting to a delayed AA flight on a separate ticket. We were pretty confident she's be able to get to the AA gate before departure, probably before boarding even began. But no seat available, so she couldn't check in, and was at risk of missing the 30 minute check-in deadline.

I kept checking while she was in the air, and eventually was able to grab her a seat, but if one hadn't opened up, she'd have been SOL.