Just wanted to note I saved The Cheesecake Shop offer a while back, and then when I went to use it the store wouldn't take AMEX. Wasn't impressed and was committed to the purchase by then as I had no backup. I checked the offer terms and there was no list of either participating or non-participating stores, so I sent the head office an email complaining, and they sent me a gift card to the value of the offer.
It certainly seems that as more businesses "take" AMEX, not all locations actually participate. I know I had an issue with my local Maccas for a while, where the app would let me pay via AMEX but if I went through drive-thru my card always failed... so I suspected the manager had turned it off at the terminal?