I did pop off an email to PC. The reply I just received requested that I call up the Any Hotel Anywhere Service Center instead. (I would have sent them an email directly first, but could not find an email address for them anywhere.)

Not that there's really anything I want to push. I've already paid the bill with my CC, and definitely don't want the hassle of having that charge transferred to the hotel card. I just would like to know if this refusal of the Any Hotel Anywhere card is a common occurrence and if there are other properties that one should know about that regularly refuse the card.

From what ksucats and chrisny2 pointed out, this could definitely be just a Vegas thing. I noticed they are very careful in this regard, asking for picture ID when I wanted to charge anything (such as a meal) to my room, for example.