You are splitting hairs where it doesn't matter. More alarmingly, by your reasoning the bank is not lending out cash either, so there is no value in considering 'cashflow' at all.

But in practice when people say they 'pay cash' for something large like a house or a car, they are usually effecting a bank transfer (well, in Korea at least). I understand that under the hood in the US this is an IOU between banks, but to the depositor there is no difference whether the money is cash or bits. What matters is whether the bank has the liquidity to provide upon request.