And provided a platinum member calls the restaurant on the list...
Cardholder isn't supposed to call the restaurant, must go through American Express.
and the table isn't available b/c it's already taken by another member, you think Amex will make the restaurant produce a table?
No, but doesn't matter what I think, American Express says:
Reservations are provided on a first-come, first-served basis, so please call as early as possible. All cancellations must be made by calling Platinum Card Concierge at least 24 hours in advance of the reservation. Otherwise, a $15 per person cancellation fee ($60 minimum, $120 maximum) will apply.