I’m actually pretty surprised OP wasn’t offered the opportunity to get on a later flight. Separately from ticket rules, Delta has a “flat tire rule” that allows airport agents the discretion to rebook late passengers onto a new flight if they narrowly miss the original booking; arriving at 6:00 AM for a 6:00 AM flight is exactly the sort of situation it’s designed for. There is no guarantee and it isn’t a published policy, so Delta wasn’t obligated to help - but I would have expected it to apply even on a BE fare as long as OP was pleasant and contrite to the airport staff, so it’s disappointing it didn’t.

OP, going back to your original question, by the letter of the ticket rules you are definitely completely out of luck. But I still think two venues are worth pursuing. 1- check your credit card you purchased the ticket with to see if you have any travel insurance that might cover the scenario, and 2- write Delta a short, polite email expressing your disappointment that the “flat tire” rule wasn’t applied in your case, using that term.

I fully expect that you will not receive a refund for the second ticket, but you may receive a few SkyMiles as a customer service gesture, which is better than nothing.