On the other hand, "private" has the downside of falling into unaccountable monopolies/duopolies. You don't have a individual choice about having a credit score, or whether banks can use it, or with which companies. You have no control, there's no accountability.

If credit scores were run by the government, then in theory democratic processes could regulate them in terms of accuracy, privacy, who was allowed to access them, for what purposes, etc. There would be actual accountability to the people, in what that there isn't when it comes to private companies.

While you say "lots of 3rd party rating services... are mainly between you and those 3rd parties", many are not. They're between one 3rd party (a bank, a landlord), and another (Equifax, Experian).

The ones that are, they're eBay, Uber, etc. Which seem more obviously defensible as being privately run.