A private blockchain allows a distributed network of arms-length peers to cooperate to come to cryptographically-verifiable consensus about who traded what and in what order - a settlement network.

A traditional centralized database doesn't easily allow irrefutability. A blockchain allows one to do this with low trust, as all of the data integrity comes from crypto.

Think of it as a database with signed tx records, and countersigned with timestamps for ordering. Any participant can assess the valid state of the world, without all of the hairy distributed database timestamping/integrity issues we all know so well.

> If your peers are trusted

Trust is never a boolean thing - and always a liability.

If your peers are trusted you could do that. Or if you can agree on a central authority you could do that.

But you are confusing trusted with known. I don't have to trust any given party in a private chain. I just have to trust the network. It's not difficult to come up with a private chain design that makes it very difficult for a malicious party to negatively impact. And you'd still have the immutability, known ordering and decentralisation amongst participants.