I don't see any point of using blockchain for private usage, just use distributed and encrypted database which already exist & AFAIK it performs better than blockchain.
Blockchain is supposed to be
decentralized ,
distributed and
immutable .Private blockchain just mean that not everybody can take part in changing or accessing the records, but it still be following all the principles of blockchain with participating nodes.
Database is still not decentralized and immutable.
It is a wrong argument. Private blockchains are neither decentralized nor immutable because they are not consensus based.
For decentralization you need free will of participants to decide and reach to a consensus about the state, otherwise it will be all about running multiple instances of same machine. Modern databases have clustering and similar features that support replication and single point of failure resistance, far more efficiently.
For immutability, you need rewrite operations to be high cost in an unfeasible level, a private blockchain/database/whatever can easily be modified by administrator and root level privilege (zero cost) because there is literally no costs involved in doing such a rewrite. A true blockchain is not immutable, it is resistant to mutation because of high costs involved. A private system just can't define any costs for nodes to operate, otherwise they didn't need any license to join.
Private blockchain is a sales discourse and a hype. It is very common in IT sector, using buzz words for marketing purposes. Corporates and organizations also love this stuff and are happy to pay for it because they can use it in PR material to tell stories about being high tech and avangard.
