If you are an owner of a Permit aircraft required to be a member of the LAA, does that mean, by being expelled as a member, you can no longer operate your aircraft?
I can't imagine that the CAA would defer the issuing of Permits To Fly *exclusively* to a private organization that has no formal public oversight but has the ability to expel their members for being, well, critical of the organization.
So if you do end up with a disagreement with the LAA that leads to your termination of membership, there has to be an alternative. And that alternative is probably the CAA itself. Less convenient and more costly, but you'd still be able to fly the aircraft.
Heck, even in communist countries you had the option of not becoming a party member.