Or is the danger primarily that anchoring will undermine your confidence and willingness to demand more (and if you felt sufficiently entitled, it wouldn't hurt you at all)?

I would guess this one. It can make you ask less, with almost zero effort on the employer's side; they don't even have to read your answer. So the cost:benefit ratio of asking you this question is huge. And even if it doesn't work on some people, it most likely does on average, so it can save a lot of money.