I can promise you they don't care. I was gold for many years. Last year, I didn't fly as much (only award travel on AA) and haven't heard squat from them except the general offer to buy back to gold.
As many have done, I've gone free agent. I just booked my first trip on United this week, in business class. Because I am in business, there are no real benefits that I am missing from elite status. The flight was ~$140 cheaper than AA and offered better flight times. The ONLY thing I am losing is about 5K of AA miles, which at the current rate of buying AA miles, I could easily get for the $140 saved, if I really want them and have the best of both worlds. (Of course, I'll now have miles on UA - time will tell if I rack up enough to make any use of them).
So no, AA doesn't miss me, and I don't miss them either. Free agent status is great.