There is no proof of work, Tezos is proof of stake. If you mean who pays the gas then it could be the state but Tezos is super cheap.

A private blockchain still is distributed across many servers. Key difference is an immutable record of what happened.

Regular databases can be distributed across many servers.

If the blockchain is private, the DMV would have unilateral control to just roll things back if they desired. All of the public mirrors would notice that but they'd have no say in it. If the public nodes did have access the DMV wouldn't be able to control the chain (and wouldn't be able to revert mass errors/failures) and they won't let things out of their control.

This is absolutely just buzzword soup.

How is that immutable if all the nodes connected to the blockchain are governed by the same entity? If you mean transparent, then ok, I can see it. However, my question then is, do we have an actual problem with transparency in car registry?

Is that a feature, or a bug?

When someone finds a bug in the smart contract which allows me to transfer the car title to you, or the DMV loses their keys, or $whatever happens and suddenly some 13-year-old elbonian hacker owns every car in California I think "restore from yesterday's backup" is a better solution than rewriting the immutable blockchain.

And, anyway, blockchains can be altered, you just need a majority to agree. Look at eth.