Bob,
your posts are almost invariably well reasoned and insightful, but this is a rare occasion where I disagree with your premise. While oversight of the actual employment of air power may require first-hand knowledge of the business of flying (so, Air Component Commander or DComOps) this is not the job of a CAS. Her (or his) role is the strategic leadership of the entire organisation and is almost always carried out in a peacetime environment where the ability to operate in a highly political, finance-driven, media-intensive environment and deliver the best long-term outcomes for a £6Bn, 30,000 strong organisations might not be related to hand-eye coordination capabilities at age 18.
That is not to say that some pilots couldn’t potentially do the job of course! However, many other branches spend a much higher proportion of their careers leading and managing people, money, infrastructure, contractors, and the public than the average pilot.
Merely as an example, we currently have a 3 star engineer in Town who many think would be an excellent CAS.