This is just their way of saying that they do not understand what went wrong. If the terms allow telephone booking you will get the money if you persist.
I booked online. What it comes down to is that amextravel / IAP codes showing the airline and not Amex on the statement, but Amex designed the offer to trigger when amextravel was the merchant. The bigger issue I have here is this seems designed to mislead people who went for the offers (or Amex is clueless). First time I’ve had to escalate and amex sent for “investigation” but the reps and supervisor both said if I don’t get email saying offer was triggered then it’s likely not going to get honored.