Outside of the phone call, they really called?, people are shocked they board early? DFW Monday AM flights? You better be there 40 min before boarding if you want to be in your group.
I frequently see them close the door and 5-10 min before departure, but I have no expectation of them pulling back immediately.
Since AA doesn't notify us when boarding starts, here is the tip. Go on the app, click on 'inbound flight', if it's at the gate more than 45 minutes before you flight is supposed to take off, you might want to be there if don't want to be 'shocked' by early boarding. Otherwise allow 15-20 minutes for boarding to start after the 45 minute mark.
Sorry it's not really that complicated.