Anecdotally, flying AA out of SFO after 9/11 was remarkable--at least until I earned status on AA. The economy check-in lines were hours long and simply wouldn't move--on busy days, you could get in line, put your bags on the floor, and that was it. When particular flight times got close, AA would send staff to walk up and down the line: "Anyone on flight xyz to JFK? xyz to JFK?" Then you would go to a separate line that actually moved, so they could get you on your flight.
Once during this time, I had a flight that was part of a long itinerary connecting to Eastern Europe. SFO-DFW-ZRH-BUD or something like that, so I *really* didn't want to miss the flight to DFW. They didn't call the flight and didn't call it. My pulse, my blood pressure, increasing. Finally, about 45 minutes before the flight, I walked over to the coordinator and pleaded my case. After a little discussion, he let me check in at Business (back when SFO had economy, business, and first check-in for AA).
So, the airline generally wants to get you on the flight. It can be helpful to say to agents, or other pax "Oh man, I am so late today, can you help me?" Sometimes they will indulge in schadenfreude, laughing evilly at your misfortune, but sometimes they'll tell you to go right ahead.
-Hayden