Unfortunately, I could see some variation of this. The clubs are often crowded and DL made the first (at the time unthinkable) move in restricting visits for card holders. Wouldn’t be a total shocker to see AA or UA follow.
Situation is a bit different. United/Chase have a card that includes UC memberships. DL allows AMEX card holders SC access, not memberships, so they can squeeze access a little more easily.