It's not about "trust" it's that keeping the accounts and books consolidated even within a single bank is problematic and between banks even more so.
Banks lose billions each year and i mean lose as they have no idea where the money went. Account consolidation within a bank is a huge enterprise.
Even within a single entity like a bank a blockchain that governs all accounts and transactions is hugely beneficial because of the consistency and the trail any action within it leaves.
Private blockchains do appear to be useful among a consortium of companies who don't completely trust each other, and don't want to pay a single company to maintain a central database.