> we trust those businesses to keep proper records, instead of putting them on a public blockchain that doesn't require us to trust anyone
Nothing really changes. You still have to trust that those private businesses will correctly use the record keeping technology. In a world with practical blockchain record keeping they would simply fail to use it or use it incorrectly.
Not if you use a contract that transfers ownership only with an on-chain payment. Even without that, you at least can't claim you lost the records, as these companies sometimes do.
Set it up legally so the loans in the system are only transferrable via on-chain payment, and teach the courts how to read those records, and it doesn't matter what the incentives are. The chain is the source of truth that way. If you're recorded as the owner on chain, you're the owner, full stop.
If you're going to setup an entire legal framework that forces businesses to use a particular record keeping system then the problem is already solved. You could legally force businesses to keep appropriate records or to participate in a centralized record keeping system operated by a trusted third party like a government or tribunal. You could also use a blockchain for this, but it doesn't offer much value.
They're already required to keep appropriate records. But the courts don't have money to go checking so they just trust companies to swear they have them, and the companies lie.
And sure, you could spend a bunch of money setting up a government database and employees to run it, but at some point it's easier to write a few lines of code and be done with it.
Nothing really changes. You still have to trust that those private businesses will correctly use the record keeping technology. In a world with practical blockchain record keeping they would simply fail to use it or use it incorrectly.