This will lead to bank runs. Why do you think citi and bofa turned them back? Because they do not have the money and the FDIC can only back 1.15 PENNIES on the dollar in the event of collapse/run....
The FDIC is not a true private insurance corporation. If it were, you might have a point. But FDIC has the deepest pockets of all. FDIC would be bailed out in such an event--probably through an emergency line of credit from Treasury.
And you'd have to assume an epic bank run to wipe out 1.15 percent of total deposits...
So in theory, you are correct. In practice, however, deposits at Citi and BofA will not be allowed to be defaulted upon.