The answer to your question is: Yes. No harm in walking up and saying "Am I too late?" Almost certainly the answer is yes, but you never know. The agent might not have run the papers down yet.

I haven't missed too many flights (other than for late inbound/missed connections), but one time last year I was dawdling in the C-gates Admirals Club at DCA to catch my "usual" 8 p.m. flight to Louisville, and I walked downstairs at 7:45 only to realize that they had moved my fate to the other pier! So I figured I'd miss the flight but I got on the shuttle anyhow, went across to the other pier, and made it to the gate at 7:55 to find the door closed. But then an agent came in from the jetbridge, saw me, asked if I was so-and-so, radioed down, scribbled something on paperwork, handed it to the FA, and let me on. So, you never know.