glad to see that you knew this problem already. One simple idea comes to mind, a lot of your colleagues did some so called familarization flights in the past. In BRU it was always very easy to just drop into the R2 office (assuming it does not exist anymore...) but I am sure that you will have luck approaching other airlines there to get some jump seat experience and see the other end of your workplace.
Pilots are more than convinced that an ATCO job is demanding and challenging at times, most ATCOs see our "office" job quiet similar. We need to pull on one string, know each others problems and ideas and it will lead to a much less stressful day for all.
I am completely convinced that any idea which will improve all our situations are most welcomed, so do not stop searching for better ways of what we do now.
The only irritating point in your posting was that the most basic knowledge of our environment appeared not to be as available as it needs to be.