I also have an issue with this and getting fed up with the so called "Platinum Service". Here is my story to add to above ...
I took up the $600 first year offer to get Platinum Charge Card in Dec last year and when I went through the Charge Card application ON THE PHONE (please note) I asked for the CREDIT CARD and they said "THIS WILL BE ISSUED AUTOMATICALLY".
Side note: Again as they other OP the ONLY reason I want the CREDIT CARD is for the free flight and I agree that AMEX could be more flexible and just give it CHARGE CARD holders. Anyway, their rules, their work and cost to issue an additional card.
Back to the story, sure enough I received the CHARGE CARD but no CREDIT CARD. So call them up and ask what's with my CREDIT CARD. They say there is no application on the system and I would have to fill out a separate form which they can send to me and it takes another 3-4 weeks to receive the card.
Needless to say at this point I told them straight up what I think of their Platinum Service!
It's just incredible to me that this company lures you into the thinking Platinum service is a whole different level and open doors for you and then they show you that this service doesn't even help you to open a few doors at their own company - how would they ever be able to cut through red tape in other situation?????