No, although real macroeconomic problems could induce a run. Simple storage without deterioration. Consumers may learn that they need resources fast. Endowment in period 0 is 1. In period 0, each consumer gets .
Banks feasibly implement the insured equilibrium.
In period 0, everyone deposits in bank. Yes - lender of last resort can substitute for deposit insurance. Eliminates runs, still allows long-term projects. With an equity cushion, this can still be low risk. Note terms of CDs.