Once you get 10k+ rep you get access to moderator tools that provide, among other things, a list of questions that have the most number of close votes. I often go through the list and see what the questions are and why people think they need to be closed, and if I agree I cast my own vote.
While I agree that users that can close (3k+) but don't have access to the moderator tools (10k+) have a harder time finding questions that are appropriate to close, I think that there are enough 10k+ mods to take care of the obvious ones. And if a [3k, 10k) user does find a question that they think needs to be moved, they can always flag it for moderator intervention or, as you said, tag it as "belongs-on-X".
In other words, I think that the system has multiple ways for people with various reputations to identify and close/move appropriate questions, that I don't think another mechanism is strictly necessary.