Thanks for all your help. Your assumption is correct: this happens with every photo I try. If I follow your instructions, when I try step 2, nothing happens. The UI is recognizing the click, because the background of the Enter Face Tag Editor button changes to a slightly darker shade during the button-down event. But the Add Face Tag button does not activate; it's perpetually grayed out and inactive.
If I then click on the "Tag faces in the selected photos" icon in the row below, and it's a photo that hasn't been tagged before, it will recognize faces in the photo, so the automatic tagging seems to be working.