AMEX can easily set its terms to disallow annual fee refunds during the 30 day grace period if member accepts a retention offer. If an annual fee posts, you get a retention offer, and you then cancel, it's AMEX responsibility how to handle that situation. It's not the customer's obligation to be worried about a corporation's P&L.
Using the Open In New Tab trick to score an extra +$1000/yr in AMEX Offers doesn't even raise an eyebrow around here. Calm down, Mary.