I'm moderately confident I know what the theorem means. I'm almost certain that our disagreement stems from different uses of the phrase "a priori dictator," and so hope that disagreement will disappear quickly.

I'm glad you've opened a discussion about this, BTW, even if it turns out you are wrong. I wondered myself what exactly the non-dictatorship criterion meant when I started reading the commentary on "Arrow's Paradox" in Ken Binmore's Playing Fair. After I read all of it I was fairly sure non-dictatorship referred to what you call an a priori dictator, but couldn't be totally sure because I didn't have the patience to sit down and puzzle through it systematically.