Earlier quoted context omitted.
That's a ridiculous argument: Banks started being a thing at the end of the Middle Ages. The Great Depression and the Great Recession were not caused by banks emerging, nor by people putting their savings in them.
Not emerging. Just being themselves with all their schemes and an economy dependent on them. A distrust of banks and their schemes at a national level might have reduced their ability to cause those problems. On top of the smaller stuff such as them delaying deposits or withdrawing stuff for bogus reasons.
Putting your savings under the mattress instead of in a bank account wouldn't have prevented the Great Depression either.
The only thing it would have accomplished is making your savings easier to steal.