A radius of 5nm is unworkable and only suits the VFR flight.
The problem isn't CAS it is the access too CAS . Come to an airfield where there is no problem with access and where the locally based GA pilots are more than happy for the airspace to be extended.
As for funnelling aircraft in to bad weather, many radar units are not even based at the airfield they work , and in the vast majority of cases are in room with limited access to windows so are unable to see any local weather, we rely on the METAR to tell us what is going on.
Whenever I give a VFR transit I always add the words, " advise if unable to maintain VMC and require to change heading or level" it is then up to the pilot to let me know if they have a problem.
Routings east/ west / north /south of the threshold work well, again at a quieter place like the one I work at you let them know the traffic they are conflicting with once visual then life becomes much easier. When you first start controlling you tend to use belts and braces in all scenarios, as you gain experience and confidence you gain a much greater awareness of what is likely to happen.
We are also not allowed to issue what are referred to as "unsafe clearances" that is why we in many cases cannot give the clearance through the overhead against a departure or arrival, once both aircraft are on the radar frequency then we can do so much more.
I have hosted many visits to my ATC unit and i know many of my colleagues do the same, it is however as easy as picking up the phone and asking if you can arrange to visit the unit , if you haven't been go along and visit if only so you can ask some of those questions that have been bugging you for some time.