Not just DL, but US, AA & UA too. If you aren't at the gate by T-(whatevere time the carrier specifies), your seat can be given to a SB pax or it can go out empty.

Given that most carriers oversell, if the flight is oversold at the cutoff, they definitely off-load the misconnecting pax.

But, to be fair, it's not just about closing the door. Closing a flight takes work by the GA, dispatcher, ramp crew, cockpit and cabin crew. That takes time. Adding one pax means redoing all of that paperwork and delaying the flight. At a busy airport, that extra 5 mins. can mushroom into a lot more.

Annoying, but it's the facts of flying.