Allowing a distributed network of untrusted peers to cooperate to come to consensus on how to update the glorified database.
If your peers are trusted, or can put behind an authenticated, authorising gateway, you don't need a blockchain - you can just use a normal database.
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.
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.
Trust is never a boolean thing - and always a liability.