Mostly agree, but missing the point that there are non-decentralized blockchains that operate like an open standard seem to failed to gain any traction (or at least awareness) since 2019, to name some: Hyperledger Fabric, Corda R3.
What does a private blockchain accomplish that an immutable ledger does not?
Nothing really. It's just one kind of immutable ledger. If you have a use case where decentralization of the infrastructure is important, you don't care as much about network effects, and you know all your counterparties, then a private chain might be suitable.