It's an open secret that money "in your bank account" isn't actually in your account waiting to be withdrawn—the banks will openly acknowledge that if asked, though it's not something they like to draw attention to in their advertising or when you're opening an account—but they aren't so open about the fact that this practice comes with investment risk, including the possibility of losing any money you've deposited with them, or at least being forced to wait longer than usual to get it back because the bank is having cash-flow issues. (Which, naturally, causes even more severe cash-flow issues for the depositors.) They have the mandatory insurance, of course, but the FDIC is more of a placebo to prevent bank runs than actual protection against systemic issues affecting many banks at the same time. The FDIC can bail out any one bank easily, but if everyone wanted their deposits back there wouldn't be nearly enough to cover the banks' obligations. The only way to avoid bank failures in that scenario would be to print more money, which would drive inflation and penalize those who put their money in less risk-prone, higher-reserve banks.
Fractional reserve banks are very honest about the fact they reuse your money until you withdraw it. What you are describing is a full-reserve bank.