There are absolutely no efficiencies to private blockchains. Are you proposing that banks will run miners merely for the purpose of passing signed messages to one another?
What Bitcoin solved was decentralized trust. I don't see any banks replacing their existing 'trust' network with code, they'll want to reverse transactions and mistakes, and will also want a veto over any distributed ledger.
Distributed ledgers are perfectly capable of reversing transactions.
Just passing signed messages around wouldn't be sufficient, you need a way of ensuring overall consensus.
Should we also include elliptic curve signatures?
But verifying the signature is part of the validation algorithm, so "validated transactions" already covers it.