They would with a 20% reserve requirement. Since that's probably not viable, maybe some changes to the interbank loan system to make emergency loans for this purpose a thing. The FDIC already has wide authority once they're called in, but a system like this could have prevented the need for it. Bigger banks could provide the money to prevent failure of an otherwise solid bank rather than be expected to help clean up the mess later.
Now, in the case of the Big 4 being run on, they are too big to fail so the Fed would just extend them unlimited funds (probably).