Missed a flight at MCI last year since the bus/shuttle took to long to get to the lot. Was flying UA with no status. Got to the gate and agent said she was paging me for the past 15 mins. She rebooked me on the next flight without any issues. I got to the gate and saw the jetway still docked at the plane. She also told me about the flat tire rule. Seats were all standby though.

Another instance, I got to SLC too late - it took me 30 mins to return a car with Avis. SLC front desk agents were very helpful and rebooked me for the next flight out the following day.

Like others have said, go to the airport as there is much more flexibility there than what phone agents can do.