Just made a non-US booking, only took 1 round of HUCA. First person I spoke to couldn't see any space, and when I mentioned that I could see availability on specific flights on Etihad.com, he told me the only reliable source of info on what AA could book was aa.com. He doubled down on that when I asked if he was sure Etihad awards are bookable on aa.com.
HUCA, got someone who knew exactly what she was doing. Searched the flights I wanted, and when she couldn't see space on one she proactively found an alternative routing.
Both people were in the UK call center and presumably on the same shift. It's kind of amazing that the quality/skill can vary so much.