Who says they don't?
I have old Exec cards from when I downgraded my Exec card to Citi's AA 'Gold' card. Same account number, etc, and even though Citi sent me replacement cards with the Gold design..I still use the old Exec card. I wonder if they would work to get entry into Admirals Club. Technically it's an active card, if they need to swipe it to check its validity...it will go through.