...why do you have to have a peer-to-peer network managing a blockchain just to verify membership data? It would be simpler to just have whoever controls memberships sign a user's public key to indicate membership (and then you add some metadata for expirations/membership levels/etc.). A user seeking to assert membership presents that certificate and proves knowledge of the private key.

Really though, I am not sure what problem is being solved here. Why do we need this in the first place? What is the attack scenario we are trying to address?