And I generally don't say my status level, though I usually lay my Executive Platinum card on the counter when asking for something peculiar where I feel they have discretion, and might say an automatic "no" without it - like "Can you tell me where on the upgrade list I am?" or "Can you tell me if my upgrade cleared?" or "Would you please put me on the standby list? even when it has a ton of people on it, or "Do you think I have any chance of getting a better seat?"
I see what you're saying, but the boarding pass does say in nice big letters EXEC PLAT, so I've found it is not necessary to show my physical card (so long as I have an AA boarding pass).