When I try it with the board window closed, the screw terminals do not appear in the list of parts.
Maybe I should try to add it to the schematic with ONLY the board window open? That makes no sense, but it's becoming clear that "making sense" isn't going to work here.
Edit: Okay, that allowed me to add it to the board (bizarrely enough...). However, when I opened the schematic window, it was not magically there - the schematic didn't change, although the little warnings window said that the board and the schematic were consistent.
It's hard to shake the nagging feeling that this is a bug. Unless I'm completely misunderstanding what this stuff is supposed to be doing, it simply shouldn't be acting this way.