must sympathize with the OP.
it really bugs me when airlines are strict about closing the check-in desk on time yet their flight leaves late! i have been denied boarding before due to being 5 minutes late for checkin, get to the gate to wait for the next flight, and my original flight is sitting there 1 hour delayed!
if we must be on time, the airline should be too. Sometimes things are out of the airline's control, and their flights get delayed. Same goes for the pax. There should be some method to override the system if a pax comes at 58 minutes.