Isn't the risk of bank runs increased without this assumption? If we're all consistently attentive and worried that we may not get our bank deposits back, then every minor hiccup at a bank could plausibly cause a run.
1. if you know your deposits are at risk then you actively work to derisk them, for example placing most of your cash in T-Bills.
2. if you know your deposits are safe then you just keep them in the bank.
The middle seems less stable.