That seems pretty insane and also this section of the policy is in the booking section. This is not in the flight check-in section. So if the ticket was purchased and the booking was confirmed and paid for then they should not have to show the credit card. Because of the purchase was made and the booking was finished they just had not checked in for their flight.

In this situation I probably would have purchased new tickets with another card and then reported credit card fraud to Delta for billing and booking and confirming a charge that they had no intention of honoring. I have a fairly decent status with Delta but I have been reconsidering using them as my primary carrier because of a number of reasons. Things like this just make it worse. They clearly don't know how to read their own policies because if there was a problem at the time of booking the booking should have said it was tentative and to bring the credit card to the ticket counter like the policy says. If nowhere as part of the purchase did it say you must present the card during check-in or prior to check in at a Delta ticket counter then I would argue that this policy doesn't apply as it's written.