HA I fly out of GSP and I could for sure see this happening, they run a tight ship. If you are checked in you can board whenever you want as long as you do not go under the 15 minute window. I would have asked for a red coat and had he/she explain to me the policy. GA making up rules has consequences. Also I always screen shot my upgraded seat number/boarding pass. I have had GAs tell me that I did not have the seat that I was upgraded to. Always have back up with Delta!
From the Delta website...
When you’re traveling within the United States, we make the suggestion to arrive at the airport at least
2 hours prior to departure. For most airports, you must be checked in at least
30 minutes before your scheduled departure time (airports requiring additional time are
listed here) Additionally, you’re required to be at the gate and ready to board
15 minutes before scheduled departure.