Seems you actually can't close/reopen until 15 reputation.
A long time ago, this didn't matter. But a while ago, we started recording close votes as flags instead of the way we'd stored them previously, which also let us start using the "recommend closure" flag for users with flagging privileges but not full close privileges.
So, this giant check for "can this user flag" includes two separate checks: one to see if you can flag, and one to see if you have close vote privileges. Problem is, if the CanFlag check fails, the whole check immediately returns false and never even attempts to look at your close privileges.
Since the CanClose check fails, you aren't shown the close button on the question or in the review queues, because you don't have permission to complete that action. But the review queues themselves don't run through the same check, which oddly shows them to you even though you can't do anything there.
Simple solution is to either rejigger the check to not fail at one step if another would still cause it to succeed, or just reassign close privileges to 15 reputation to match the flag privileges and prevent confusion and just forget about the weird code thing.