In the "Menu Layout" preferences item, I navigated to my top menu, the one I want to open up on long menu button press. I was rearranging the menu items, and when I moved one of the items ("Audio & Video"), it uncovered another item ("Behaviors") which seemed to be hidden underneath the "Audio & Video" item. So I deleted the "Behaviors" item, and my menu operation now works correctly!

I couldn't believe that I saw that correctly, but I went to my second computer, which had the exact same problem (because I had duplicated preferences between machines), performed the same procedure, and it fixed that machine too.

Weird. But apparently ok now (fingers crossed)...