Okay Bank lending is one aspect of a Bank, which seems doable with some caveats about size of loan and trust but not the only one. The other two personal banking areas are saving and current accounts.

So what does that mean in a Bitcoin bank?

A saving account traditional holds your money in trust returning you an interest rate bonus for doing so. The interest is generated via the bank being able to use your held money for loans, investments etc. The problems are saving guarantees, currently most countries guarantee consumer savings up to some value even if the bank fails, that of course can't happen for bitcoins, so do would we need a guarantee? and if so how would it be achieved?

My approach would be that saver specifies how much of their savings can be used for interest, from this the amount of interest s generated but also is potential lose if Bank closes.
So I save 10BTC and decide to allow 5BTC to be used for a guaranteed interest rate return of 1% per month. So assuming no component interest (could be the savers choice, which pool interest goes into), I earn 0.5BTC interest per month.
A Bank may set a minimum amount to disallow free savers (0 used for interest), on the other hand it might just let it happen (free saving account) as a goodwill / PR thing. I'd also suggest that a 0% interest saver account is really a wallet backup account and may be clearly to call it a backup account rather than savers to show it earns nothing.

A current account, the most used part of a current account is paying bills.

For a BTC current account i'd suggest

Instant transfer internal to the bank and to other banks with a data transfer agreement (A Bank could fast track bitcoin transaction time because it knows both ends, if they are both customers)

Traditional bank to bitcoin bank transfers - probably local country only but transferring money from Cash Bank to and from bitcoin
bank would be safe and cheap for the Banker.

Statements and tracking - allowing you safely see what you have, where you spent it

Secure - you wallet and bitcoins safely stored and protected from local effects

Overdraft - A bank may choose to allow a customer so head room, a predefined loan space...

Fee - A fixed monthly charge would seem to be the simplest and fairest system. The customer pays a small fee per month for the current account to be maintained.

Would anybody find such a thing actually useful?