You can still determine the log likelihood of your predictions across the whole set of questions, even if you only have a single sampled response for each question.
Yes, and it can be simpler to calculate it, but it tends to have worse theoretical properties. In particular, with log likelihood we can sum the scores for a sequence of conditional probability estimates and we get the same score as if it were a single estimate for a single compound random variable.
Brier score not only lacks this property, but you don't even necessarily maintain the ordering of such scores.
