The FDIC is an agency of the Federal Government. In reality, it can never go broke. All FDIC insured accounts will get their money back in case of a bank failure. That's the whole point

"If you promise people their money, but don't actually have enough money to pay them (if only 10% of customers close their accounts and cash-out, the bank is done-for) then you're running a ponzi-scheme."

No, that is not by any means a ponzi-scheme. Not being able to pay out deposits due to the normal course of business does not constitute a ponzi-scheme. It would only be a ponzi scheme if the bank was originally setup to take deposits, with no intention of ever returning them, or ever paying interest on them as advertised.