Depositors can be made whole without receiving little green pieces of paper.

Checks, account transfers, wire transfers, etc.

There is no fundamental limit on the Federal Reserve to create (or destroy) money as needed. The Fed, as other central banks, does however exercise that power very judiciously, and with specific targets (inflation, unemployment) as its foundational charter.

Note that both inflation and unemployment are not assessed by the Fed, but by an independent federal department, Labour. It's a classic instance of not giving a single entity control over both the means of control and the measurement of success.

> The bank makes a business claim that they can redeem ALL of the green paper back to everyone at the same time.

They never claimed this. Everyone has known about bank runs, fractional reserve banking, etc. for a long time.