Much as Bitcoin addresses the double-spending problem to ensure that a single output isn't spent twice, a "credit interaction" would be added to the ledger as a continuation of an existing history (or a new one). The zero-knowledge proof would assert various attributes of that history.

At one point or another, the history could be associated with a specific human. Or not; depending on the goal of the design, a single credit history might pertain to one person, multiple people, or a business association. A single person might have multiple histories -- though longer-lived and more active ones would be more valuable, which reduces the viability of a Sybil attack.

Loss of private keys is a problem if users are their sole maintainers. A credit union would surely be willing to safely store a key for a 2-of-3 transaction that empowers it to manage a customer's history.

Distributed ledgers absolutely do not need miners, especially if level of trust is a variable aspect of the design.

By the way, your initial question could have been interpreted as a request to engage in a brainstorming discussion to produce a blockchain-based public record that preserves privacy. If so, you should know that your subsequent response is inconsistent with that sentiment, though I'm giving you the benefit of the doubt by continuing in the spirit of additive discussion. If you're not interested in brainstorming, then please state your intentions so we don't waste time talking past each other.