All of this still requires a central authority to decide these things, so they should just manage the database themselves. The "public" would not be permitted to just write to this permisionlessly.
And, yes, in a lot of cases, it makes more sense for one entity to be the sole arbiter of truth.
Who is the arbiter of truth in private/business blockchain? Public blockchains that produce currencies reward operators of nodes with a token that is self-contained within the system. How does any of this map to private/business chains, or any other "blockchain" that is somehow mapping some truth state external to the chain into it? It seems these words are thrown around without any meaning, for marketing purposes.
The two only share some characteristics.
It's sort of unfortunate that blockchain was ever used as a term for business distributed ledgers. The Hyperledger Foundation has a ton of material online. (As do many others.) Proof of work is not used. The ledger is typically not open to the public or only to a limited degree. There are various other consensus mechanisms depending upon the requirements.
Mind you I'm actually not sure what the CA DMV would gain. It seems they're the only source of truth that matters with respect to CA residents. Title information maybe given that does move from owner to owner.