Because the primary card holder's SM # is tied to the credit card in the computer system, and that is who must be one of the travelers my guess would be that no you would not have to show the card.

Case in point is as a primary cardholder try to make a reservation while not logged into DL.com and you are not given the option to "pay with miles" but once you are logged in you are given the option to "pay with miles".

The system, at least at this point is smart enough to know you are a DL Amex cardholder. Also Reserve cardholders have had that printed on their boarding passes so that is another indication that TA's and GA's systems show this information.