I am inclined to side with the DL GA.

The GE card is indeed a valid ID. However, both TSA and DL say the list of the approved IDs are for "travel through security", but never mention check-in.

FWIW - a passenger can enter airside with a valid BP and TSA approved ID. But the GA can in fact refuse check-in because the passenger presenting incorrect ID. For example, one can use the GE card to airside with a BP for international flight, but the GA will not issue a BP without the passport.

I know I am playing with words. Some of you may not agree. But indeed - DL never spells out what kind of documentation is acceptable for check-in. On that basis, I side with the DL GA.

I agree it is highly unusual. But I respect OP for not doing so, as it is his/her right, as well as trouble if occur.