My understanding is that there is one time to remove the elite flyer tags from your bags - when the airline baggage crews are on work slowdown or headed for a strike. They theory I've heard is that would be precisely the bag most likely to get misrouted or lost, as at that moment, they are looking to leverage some upset that would be transmitted to management.
Nope, this is not gospel, but I was told that by somene on the insde, some time ago, and it makes sense, even if only a small percentage of handlers would resort to this (and it would probably be in MIAMI, he said, displaying his bias!)
It is funny - AA won't sell GOLD or PLATINUM baggage tags, but you can order new Admirals Club tags! Uh, wouldn't the same tag manufacturer be happy to make a profit, and wouldn't a like profit accrue to AA that I am quite sure the AC provides on tag sales? Now, THAT'S logical marketing!