All
I have read most of the thread and haven’t seen if someone found a solution and posted it, but sorry in advance if this is a repeat.
First, thanks for all sharing because I’ve been having the same issue and never realized it was Delta-specific. Now I know!
So I found a very simple solution: check in via a desktop and they automatically will ask you after you enter passport information what is your status (permanent resident, tourist etc...).
Upon checking the box “permanent resident” you will be prompted to enter your green card number and VOILA! Got my boarding pass - and a weight lifted off my shoulders. I never like to get to the airport early and I never check bags so the idea of potentially having to stand in line for some time in case the kiosks don’t work or anything else is always a dread. Hope this will work for everyone else!