This could probably be set up through advanced permissions. The add-on you are using only creates shortcuts to those places within the dashboard, and doesn't override any privileges.

The bigger question is, are you giving "anyone" the option to move pages, or specific people that are running the website? In that case you can turn on advanced permissions, create a group, add whomever you need to be able to move pages in the sitemap to that group, and set the permission on that group to be able to do that.

http://www.concrete5.org/documentation/using-concrete5/dashboard/sy...

Follow the links beyond that on assigning permissions.