Like I said, “apples and oranges”.Sorry, but wrong.
1) Post Office is just another cargo shipper. They do not have the authority to hold a flight (if they did, we would have had a delay code for them). If they don't have their shipment ready by the cargo cutoff time, it may not go.
2) Plane being full is not a factor. Passengers have both a check-in time cutoff and an at the gate ready to board cutoff. Miss them and your reservation can be cancelled. These days, those times are typically 45 minutes and 15 minutes respectively. So at 15 minutes before scheduled departure, if the plane is loaded and there is no one in the gate room, close the door and send the plane on its way. But note that between closing the door and first movement, there are still several things that have to happen. And scheduled departure time is when the plane is supposed to be moving, not just preparing to move.
3) Basically true. But Ramp ("below the wing") does their thing separately from what Customer Service ("above the wing") is doing and the passenger loading door may be closed even while Ramp is still doing their thing with bags and cargo.
4) Better to leave ATC out of it. Normally, they are not involved until the flight crew calls for push-back or taxi clearance which won't happen until the plane is ready to depart (so from the perspective of a late arriving passenger, the door is closed so the flight has already departed, even if the plane is still there).
Unlike how railroads traditionally operated, the published schedule provides no movement authority (even the public railroad timetable provided no movement authority, only the employee timetable did). If a flight is ready to depart early, there is no one who is going to say "uh, you're early, you can't leave yet"). So once all passengers, bags, and cargo that were ready by the appropriate cut-off times are on-board, there's no reason not to send the plane on its way regardless of what the schedule says.
The only train that operates anything like that, is the Auto Train...