Granted, I lack the experience of other long-term members of the cabinet and other high government positions, but from my understanding Cabinet Ministers and their advisors make up a large portion of those channels. It also feels weird to say a Cabinet Minister is not a member of the “Executive” (as you said before, “As I understand it, a member of the Executive is someone serving in an executive branch position other than the Cabinet”), but their advisory board is.
Between your latest post and the one before, there appears to be some sort of discrepancy, as you say, “members of the executive branch (i.e., the Prime Minister and Cabinet ministers)”, which contradicts your previous statement.
Maybe I simply misunderstood what you mean by the “Executive”, but it seems strange to separate the branch into the “Executive” and the Executive, if that’s indeed how you see it, and it’s something I personally don’t do.