Your understanding is incorrect. What you say above is that if you pay for the checked bag fee with the Citi AA card, then you don't end up actually having to pay for the charge. The way it works is that if you have the Citi AA card and it's properly associated with your AAdvantage account and your AAdvantage number is on your reservation, then you are not prompted to pay for checked bags at the airport in the first place. If they ask you to pay, you've already lost the battle at that point, regardless of the card you use to pay for the checked bags.

You should approach it in that manner. It's not that you failed to get automatic credit for your checked bag fees - it's that the airport agents failed to give you the free checked bags benefit.

Step 1: Find out why that happened. There is a reason, somewhere. Perhaps Citibank didn't associate your credit card with your existing AAdvantage account, and instead opened a new one? Whatever the reason is, find out.

Step 2: After you know what happened, hopefully part of that will be to establish that you were actually eligible for the benefit. At that point, approach AA customer service (not regular reservations) to see what they can do about it.

Further advice: follow Step 1 and Step 2 first. When you're on the phone with AA, don't use terms like "False Advertising" or "promised" or "Consumer protection agency" and try to not be emotional about it.