We had a similar situation last September on an LAX-LHR flight. When online checkin said we had to check in at the airport, we got to LAX four hours early, Which at first seemed crazy, but then absolutely no one could check us in. At least six different agents (plus the support desk) worked over two hours to no avail, including our getting a “your reservation has been cancelled, but your flight credit is safe” email in the middle of it all. As with the OP, everyone could not have been nicer, but it was super stressful. Finally at about 85 mins prior to departure, a new agent was brought in who fixed things in about 10 minutes and handed us boarding passes (original J seat assignments lost but new J seats were fine). The unsatisfying part is that I asked her what the issue had been and she said it was too complicated to explain. I was so eager to get through security and to the flight that I left it at that, so I still don’t know what went wrong,
Anyway, we did make our flight so I don’t have anything to offer regarding compensation, but wanted to acknowledge that you’re not alone with AA IT.