The question of whether moderators should mark an answer as accepted is moot, because they don't have that capability.
But actually, no, they shouldn't do that anyway. The acceptance is a mark that the answer helped the asker. If no answers succeeded sufficiently well, then it's not up to moderators to decide that they did.
It is true that old questions can be bumped to the front page if no answer has been accepted and no answer has been upvoted. I have often thought that it would be good to stop that happening, and a mod "don't bump" checkbox would be nice. But another way is to upvote an answer. If no answer is worth upvoting then I suppose it's arguable that a better answer should be provided, and the bump is a means of getting one.
However, it's certainly not up to moderators to decide that an answer should be accepted, and even less so if the only available answers aren't all that good.