Good information in this thread, but just to summarize since it is a bit spread out, if you have a Reserve Card, there are two options for secondary cards:
1. You can add a Platinum Delta SkyMiles card for no cost, it looks like
this. This card does not provide any benefits to the person named on the card, I find it useful for my wife so that her spend counts towards my total spend to earn bonus MQM.
2. You can add a Delta SkyMiles Reserve card for $170, it looks like
this. The person named on this card does have access to the SkyClub.
However, the holder of a second Reserve card does not have that linked to their SkyMiles account, so what this means is that while someone who is the primary Reserve card member only needs to give the SkyClub their boarding pass, someone with a secondary card will need to give them their boarding pass and the credit card, much like someone using the American Express Platinum card would.
They were incorrect, I think the best solution if you couldn't or didn't want to push it at the time is to charge your entry to that same card then take it up with American Express and/or Delta after the fact. However, I think the simple solution would be to ask them to refer to their chart of cards that provides access, which should clearly show the card you are holding.
Edited to add: Does anyone have a picture/scan of the card that they have at lounge entrances showing the different AMEX cards and which ones provide access and which ones don't? I couldn't find it using Google ...