I think we're talking about different senses of the word "trust". I'm more referring to it in the sense of "I trust you to competently administer this database and not tamper with its data" rather than in the sense of "I believe the the data you're committing to be accurate".
The former is what a (public) distributed ledger solves; a central authority is problematic even in interstate contexts, let alone international. Blockchains don't directly solve the latter, but do provide tools for tackling it - namely, by making the submitted data trivial to audit. Using your example:
> If you wanna get a good deal on a used car being sold, just submit a fake entry about it having been totaled in an accident. Instant discount!
Setting aside the immediate footgun here (since you've nerfed whatever resale value you'd hope to get out of it), your single datapoint is less believable than the concurrent agreement of:
- A police department noting "a vehicle with this VIN was involved in an accident"
- An insurance company noting "we paid out an insurance claim for a vehicle with this VIN"
- A towing company noting "we towed a vehicle with this VIN to a repair shop"
- A repair shop noting "we serviced a vehicle with this VIN"
In short: there are a lot more opportunities for verifying information when it's on a public ledger than there are when it's stuck in centralized databases.
> And not to mention all of the PII that’s core to DMV’s operations that can’t be public, so it can’t be on a public blockchain anyway.
Nobody said literally everything the DMV does should be on a blockchain.
In this context of "CarFax but better", very little (if any) of that PII needs to be present in the ledger - and certainly not in plaintext. The core datum here is the VIN, and that's already far from private.