I do not understand why one would fetch stacks of paper from the bank if one believes toilet paper is the more valuable type of paper you should hoard and maybe also trade with.
Seriously, why would the end of fiat cause a bank run? I would expect it would cause a switch back to trading physical goods directly, or the emergence of another currency-like thing or good (like cigarettes might have been in the past).
When bank runs were common it was normal for a bank to go out of business while nothing really happened to devalue the currency so you'd essentially be racing other customers to withdraw first. The Fed's guarantee to cover accounts in the event of a bank failing stopped bank runs entirely.
Because fiat is created in 2 ways: central bank and bank. Cash in hand means it's central bank money, numbers on your bank account are the multiplied variant. Both are part of the fiat system.
Seriously, why would the end of fiat cause a bank run? I would expect it would cause a switch back to trading physical goods directly, or the emergence of another currency-like thing or good (like cigarettes might have been in the past).