I spoke again with DL's online support desk about the OLCI problem.
After being given sloppy answers like:
“We have problems with OLCI for some of our customers; please call us back iff this happens again”,
I finally spoke with a supervisor who took the time to call the web engineers and gave me a definitive answer: the green card module that would allow us to check in online doesn’t exist on the Delta.com website at all. He promised he would escalate the complaint and give me feedback as early as tomorrow... I pointed out that not only AF/KLM does have those fields to enger GC information but that AA does too… Hopefully they’re responsive to that.

If you're having this problem too, please call 888-750-3284 ASAP and specifically asked to talk to a supervisor, in order to (keyword) "escalate" the complaint. If it's many of us, maybe they'll actually address the issue.