That does sound odd. Now I’m curious: do you have a cite? Who are these people who are trying to claim that the “carrot dangling from a stick” interpretation simply does not exist? I don’t think I see any of them in this thread.
It should be “the carrot or the stick”, IOW are you going to tempt them to do the right thing or whip their butt into doing it. Funny you should mention a carrot on a stick, though, as I remember a cartoon of a donkey with a straw hat on, with a carrot dangling off a stick forever in front of him, and a human behind him brandishing a stick.