There's a dozen reasons why an aircraft could "just sit" there after door closes. Since this was a small airport, I'm betting this wasn't the ground crew. Maybe there was an issue with the final weights and performance. Maybe the wind shifted and the takeoff runway was changed. This happened to me just last night WHILE we were pushing back. It takes a few minutes to reprogram the FMC and double check the thrust numbers and V-speeds. We also have to go over the new departure procedure and single engine procedures if there are any. It could have been an ATC reroute. That also takes 5 or 10 minutes to reprogram the FMC and make sure there's a enough fuel on board. There could have been last minute maintenance paperwork to finish up. If its something minor and the thing is fixed or MEL'd we can still close the door to get things going while waiting on maintenance control to send the final sign off via ACARS or over the phone. They could have also been waiting for a EDCT time (wheels up) time issued by ATC. This is pretty common for going into a hub and if at a small airport its certainly more prudent to not run the engines if you have to.
Once airborne, ATC has no idea if you're late or not. But the pilots will try to go faster, (burning more fuel) and they will ask for shortcuts. This can save a few minutes, but thats about it. If available they can ask for a closer runway, but that is not always available depending on how busy it is. DFW, in general, arrivals coming from the east will last on the east complex, west arrivals on the west side, regardless of the terminal. If it's really busy, there's no changing sides. If not busy, ATC will often accommodate aircraft parking on the opposite complex.
Once on the ground, its difficult to prioritize late aircraft. About the only thing that the airline itself can do is to make sure there's an open gate at the hub, and they already do this anyway. Ramp controllers will hold the push of one aircraft if another is coming in behind it, for example.
Summary. They do trying to prioritize late flights, but its not always possible.