I sympathize with the travelers in the video as I have been in that same situation and done that myself. Last October I booked a PM ORD-PHX-MIA-SJU, reason for routing is that it gave me a late departure from ORD with an AM arrival to SJU without any need of taking time off or sleeping at MIA, in the spirit of MR, the added stop at PHX was cheaper and added more miles than just going ORD-MIA-SJU.
On our way to PHX, a sandstorm had our flight diverted to TUS, I called the EXP desk regarding our connection, originally 90+ minutes, now, ???. PHX was shut down due to the sandstorm and I was guaranteed that I would not have any problems making our connection. Our plane is refueled and we take off to PHX, as we land I checked my mobile to see that our departing MIA flight is on the gate next to our arrival flight with a departure time of 30 minutes, we immediately got out of the plane and went to gate, only to find a GA walking out of the jet-bridge and closing the door. She refused to board us, there where 7 people on the ORD flight that where trying to make the connection. I could not believe it, they knew they had that many travelers waiting to connect, plane was arriving and landing on the gate next door, this is the last flight of the night. We start banging on the window, the jet-bridge is still connected to the plane, I can clearly see the pilot, I called the EXP desk and mention our situation, agent tells that the plane shows a departure in 20 minutes, doors shouldn't be closed as boarding is still available. She puts me on hold, at this time I noticed that the GA receives a call, and also the pilot appears to be talking to someone as he is also looking towards the gate, the GA reluctantly opens the jet-bridge and scan out tickets. I settle into my seat and I clearly see one of the ramp agents walking with my checked suitcase from the ORD flight towards the MIA flight and place it on the belt for the plane.
We end up leaving 15 minutes after the original scheduled departure time, and arrived MIA on time and continued to SJU. Yes, this happens and it is rather frustrating situation. Not everyone is prepared for an unexpected layover, but if anything can be done to avoid one, you try. I was lucky that the EXP agent help me out and I sent an email thanking them.