> blockchain to handle DB + access api for relatively slow db operations I think is one of the rare great current blockchain use-cases

Ok, sure, it's OK for car registration to be slow and expensive in compute terms.

But what benefit does blockchain have here? What could blockchain possibly add that a private database maintained by the state with otherwise equivalent functionality couldn't do better?

This is a textbook situation where there is a trusted third party. In fact, the state is the party whose opinion controls in this situation. It's not like people are registering cars for fun, the are doing it because it's a government requirement.