But you wouldn't be able to authenticate the bank transfers- or messages- as real. It's the exactly same situation the so-called cryptocurrencies run in to, and why all three have added signatures onto it. That doesn't make any of the three cryptography- just something that exists better with the support of cryptography.

A cryptocurrency can exist without the signature- it's less useful, but 'just trust me bro' is basically THE underpinning for first currencies since the beginning, and the source of a lot of the problems with them.