You're right that the private vs. public is a very important distinction here.
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.
Your "in theory" is doing a lot of work there. So much work. Have you heard about no-fly lists? The latest ICE actions? The Red Scare? Giving the government MORE power is almost never the answer.
> Giving the government MORE power is almost never the answer.
I've also heard of food safety regulation, airline safety, public schools, libraries, science funding, workplace safety regulation, building safety regulation, the list goes on.
Giving the government more power is quite often the answer. Sometimes it's the best solution, sometimes it isn't. But it's definitely not "almost never", that much we can be sure of.