I don't believe that controllers would consider only doing half the job. I don't know any controller that would not do whatever they feel necessary to ensure safe passage to all aircraft 'on their frequency'.
Outside of Controlled airspace, thats the crucial difference at a procedural control unit, the controller can only provide seperation or traffic on or to known aircraft.
One aspect that I personally feel should be improved is the criteria that a CTR is provided. It is my understanding it goe's on Pax numbers rather than aircraft movements and types.
