A permissioned blockchain has a gatekeeper, just like a central database. So what actually distinguishes it from a plain database?

The gatekeeper controls who may write, but no single writer controls the state — the permissioned writers still don't have to trust each other, and the ledger is integrity-protected across them.

This is the paper's central tension. A permissioned blockchain shares a lot with a centralized DB: known participants, a central admission authority. The difference is trust among the writers. In a plain shared database, the writers must trust each other (and the operator) not to tamper with the books. In a permissioned blockchain, admission is gated but the writers can still be mutually distrusting — consensus, not trust, keeps the shared state honest. Remove the mutual distrust (all writers trust each other) and the blockchain adds nothing over a shared database — which is exactly the branch the methodology sends to "no blockchain."