A custom role cannot see the Run Unrestricted checkbox under Results subtab when creating a saved search.The Run Unrestricted feature on Saved Search is currently available for Administrators only.
System Notes Search Returns No Results for Non Admin Role When Run Unrestricted = T
A non-administrator user is unable to see User Notes and System Notes created or set by others when viewing a System Notes Saved Search Results even when the search results criteria is set as Run Unrestricted = T.
As per design, the System Notes Saved Search Results are filtered by the currently logged in user when the role is non-Administrator. It is the System Notes records that are being filtered, and not who owns the Saved Search.