I'm saying something about currencies because arguably it is the only legitimate use-case for "block chains" as popularly understood. How is this hypothetical

I'm saying something about currencies because arguably it is the only legitimate use-case for "block chains" as popularly understood. How is this hypothetical non-currency decentralized ledger written to? Who has rights to do so? And how does the rest of the network verify that what any individual writes to it is valid? Open public ledgers work because they reward actors (with a currency, used and existing inside the network itself) for securing the network and validating transactions. How does a non-currency ledger reward people for running nodes? How are transactions validated..proof of work? How is consensus reached? If one reads the ledger, how do you verify what it says matches the real state of the physical world it supposedly represents? If you're saying this is none of these things, then why call it "block chain", vs just a permissioned database, centrally controlled?