Sure, but you’re making the assumption that no bank is a bad actor, which is the thing blockchains solve, you don’t need a Blockchain to circle money around between your friends, but it is needed when you introduce random persons who can screw the system.

For example, let’s assume that you’re using public/private keys to ensure that only one bank can sign their own transaction and it’s easily verifiable, let’s also assume your id is the transaction hash which consists of sender, recipient, value and datetime (otherwise it’s super easy to send the same transaction with different values). Now I’m a rogue bank, how can I screw that system? Simple, I need to pay bank B for something, so I send him a transaction for it and at the same time I send every other bank that I’m draining my account to my friends bank C. B thinks I paid him, but every other bank thinks I gave my money to C, if B tries to spend it he will figure out that other banks don’t recognize that transaction, and even if he shows the transaction they also see the other one made to bank C with an earlier time which bank B never saw, which means I never actually paid bank B, even though B thought I did.