I personally see the question of "how should OWL work/process votes" as a distinct question from "should OWL be a standalone Ministry/its leadership be part of the elected Cabinet".

OWL can be its own Ministry and still follow the exact same process it currently does. OWL could also have its process wildly altered after discussion by the Direction and their Staff and not be an equal Ministry seat. 

Perhaps an elected Ministry portfolio could be a mechanism by which to induce change to OWL and accountability for its structure and leadership. These are certainly interrelated questions, but they are not the same, and I'd hate for the response to the questions raised by the original post or the discussion points in the Cabinet thread to simply be "make it a Ministry".