It should. Here's what I do. (See? I'm keeping my promise!)
Walk the Board
Follow along with how this works via my previous blog post.
Now fold in the following.
There are normally more items on the task board than there are people. No surprise here. This means that it would take more time to trivially answer the questions of, "Any impediments?" if you do so per backlog item versus per person. So while I believe it's better practice for a conversation around an item to be brought fully to done (for the purposes of the Daily Scrum), I usually ask the impediments question after the board is fully walked.
Over time, I've found this is the start of a different section of the meeting... one where we address wrapping-up questions, asked of, and answered by the whole team.
Anybody working on things not currently captured by the board?
Anybody unduly distracted?
Any bugs with heat?