Just pointing out that this seems to be your own interpretation and isn't held in any legal doctrine I've been able to find.
In-fact it doesn't have a lot of historical or academic backing either: Historically, the separation of church and state was about removing the special benefits of state-sanctioned religions so that other churches could exist.
That was explicitly about the legally registered organisation, and you can see this now in how legally registered churches are constantly trying to find ways to legally divorce themselves from linked entities so those entities can receive state funding. That is 100% about the legally registered organisations.