Since I currently work in baggage for a major US carrier, I think I have some experience with this. Arriving three hours early does not give you a better chance of having your bags loaded than if you arrive one hour early. The bags just go down the chute and are are hopefully grouped together with other bags for your flight, until ramp is ready to load the bags. And, just because you checked more than one bag together, does not mean they all will be loaded onto the flight; they are not velcroed together. Your bag is not loaded for an international flight until you actually board the flight.
ALL airlines to my knowledge, at least in the US, close the door to the jetbridge 10-15 minutes before the flight for domestic flights, usually earlier for international. So, pax who run up to the gate at 13:50 for a flight that leaves at 14:00 are usually out-of-luck. Pleading for the gate to be opened is a waste of time. You are late and your seat has probably already been released and assigned to a standby passenger. It is not a simple issue to "just open the door, I can see the plane is still sitting there."
The gate agent can see who is arriving on connecting flights, and believe me, this is being tracked. We don't want to prevent you from boarding, but when you run up to the gate after the jetbridge door has been closed, and I can see your connecting flight arrived 30 min ago, two gates down, and you are loaded down with Starbucks cups, I honestly don't have much sympathy.