is there any other Christian denomination that is secretive, institutionally hidden from view, and that has a public face, and a private face?
Yes: all of them. The technical term is "politics".
I wasn’t being obtuse, your post just wasn’t very clear.
I’ve never noticed what you just claimed, regular churches seem open and non-mysterious, and without a walled off base somewhere that lives in secrecy and isolation, and which has it’s own private operations within the institution and it’s permanent leadership elites, and then a public face that it presents outside the walls.
I know that Mormonism does that, no one really knows what goes on with it’s Pope and the inner council of the holiest, and it has institutional secrecy and history, we know that there is a level of division between the deep, inner Mormon church, and regular Mormons and the public.