The Bank’s advantage (and hence its value) is pooling large numbers of diverse individual lending/saving requirements. The statistical stability of a large diverse pool allows the bank the borrow short (normally cheaper) and lend long – their business model.

The short vs long balance is normally stable enough that banks can operate on small fractional reserves – leverage the business model.

Having a large pool (many diverse exposures) at each point in time neutralises the temporal issue far more effectively than small groups of individuals would be able to – hence the banks add value by being market makers and by providing a stable interest rate structure.

As long as the pool is large and diverse there should be negligible chance of a liquidity crisis for a bank, because all transactions in the economy have two parties and so constitute a net inflow and equivalent net outflow to the banking system as a whole.

But as you have extensively commented, liquidity crises can and do occur. A statistical slant is that the “independence” property of the diverse pool evaporates as each successive calling up of short term funds increases the chance that others will do likewise. In reality it can be more like the madness of crowds! But even such a run of fund outflows must be net inflows elsewhere.