The problem is the contract of carriage with AA specifies things like you must be “at the Gate” 30 minutes before departure. It then says boarding ends 15 minutes before departure. Both these times are immutable, at least until they publish a delay, and perhaps even then still (haven’t dug that deep).

The boarding ends at T15 doesn’t mean they can’t reassign your seat any earlier than, they can reassign your seat at T30 (T45 if you haven’t checked in) if you don’t respond to a call to identify yourself, taken in a reasonable manner or if the other passengers have boarded and no one is left in line, even before T15 (They would HAVE to reassign the seat before T15 in order to end boarding and close the door at T15 {a legal fiction, but in the terms, etc.}).

Your fig leaf is the Agent’s comments. Stay focused upon that while dealing with AA, and concentrate on it and you might get a refund and a few miles, but there is no way AA will accept this was an IDB and somehow compensate. Unfortunately, the recent TikTok trend on “airport theory” (arriving at the airport at Boarding time) has likely turned Corporate attention on this field, and weakened your chances.