Flaw:

“When a bank fails, a quantity of money equal to the bank’s deposits disappears from the economy.”

Truth:

Even without deposit insurance, money, in the form of bank credit, would not disappear when a bank failed. Depositors, who would comprise the largest group of creditors, would take action to protect their claims. They would probably take possession of the bank's assets and sell them to another bank for new account balances.

New money would replace old money. Depositors might take some loss for the amount by which the sale of assets falls short of deposit liabilities. But no wholesale loss would occur.

With the deposit insurance the same action occurs under the direction of FDIC. In that case the FDIC would cover any shortfall.