"Flight closed" is not about closing the doors (jet bridge or aircraft cabin), but about the administrative process which goes with a flight departure. Closing the doors is one part of all of that, but in order for an aircraft to push on time, several pieces of data, including passenger count, must come from the GA. That cannot happen until the flight is closed. This is not to suggest that the flight cannot be reopened, only that this is relatively rare and is done through Operations, not the GA's discretion.
AA's passenger deadline for domestic is T-15. At that point, AA may offload those who are not boarded(ing) and fill the now empty seats with standbys or simply send the aircraft out as soon as it is ready to push.
All of that said, if you are actually close to the gate and see "flight closed" and there is still a GA present, I would still head to the GA and see what can be done. You can't know until you have tried that it won't work. Most especially true if the aircraft has taken a significant delay and there are seats.