I travel with a trained service dog, and I've only once had a minimal problem with American. In fact, their "SAC" desk for passengers with disabilities tends to be terrific. They're almost always pleasant and helpful. I also had a United flight cancelled and was endorsed over to American. Even though I didn't have AA-specific paperwork, the AA agent made absolutely sure that everything would go smoothly. They really seem to care.
Once you send your paperwork to the SAC desk in Dallas, they will review it and issue a Service Animal ID. You just enter this in to your reservation each time, and it saves a lot of effort. The ground crew can see that you have everything approved. Moreover, you can see yourself when you view your reservation online.
The only time I had it rough was a few weeks ago at LAX. I was checking in at T4 and it was packed, so the lobby coordinator asked if I wouldn't mind walking over to T5, which was empty. The agents were clearly livid because they were enjoying not doing any work. The agent assigned to me said, "And what is your disability?" I replied, "If I'm not mistaken, you're not allowed to ask me such question." She shoved a boarding pass at me and rolled her eyes. I'm sure she has a charming existence. Milk of human kindness.
As for seating, I usually aim for the forward cabin, so that resolves any issues about space and boarding. My dog is quite small, but I find that a bulkhead window is the best place.
As for cabin crew, some barely notice, others have made a point to stop by and say hello. Since the pandemic, I think they're too afraid to get too close to a dog. (Officially, one is not supposed to pet service animals--but service animals can still be cute, and they still like being petted.)
American does a remarkably better job than their Dallas neighbors at Southwest. I've had lectures, inappropriate questions, snap judgements from Southwest ground crew. I also have watched more passengers try to scam Southwest (a woman at LAX yesterday just started yelling out her son's potential diagnoses to see if one would be okay. "He has ADHD." "He has ADHD!" he needs his emotional support animal." So perhaps the Southwest staff have had their fill pet shenanigans. I just don't feel that I need to listen to their lectures about other passengers' animals.
This was a big scene in the check in hall. When I (recently) saw a woman claim a faux service animal at American, they handled everything with great dignity and respect. The agents spoke in honest but positive tones, allowed her (and the dog) to fly as a pet, but kept everything in a polite tone of voice and without threats.
I will give Southwest one big award for outstanding treatment toward animals. Pre-pandemic, a flight attendant stopped by after the safety demo and reminded me that, "If the masks drop, put on your mask, then put one on him." It is the right thing to do, and I was impressed that she took the extra mile to remind me.