But that's a poor definition of a blockchain. A blockchain is merely a distributed ledger with certain properties from cryptography.
If you spin up a private bitcoin network, it's a blockchain even if nobody else knows or cares about it. Now, are non-open blockchains at all useful? I suspect so, but I don't know of any great examples.
The wide space between 'membership is determined in advance' and 'literally anyone can make a million identities at a whim' is worth exploring, IMO.