A.k.a. “moral hazard”. When people take financial risks (such as holding more than $250K in FDIC insured bank account), why should taxpayers cover their losses?
Banks that may be otherwise solvent may become insolvent because of a domino risk of bank runs.
Probably the right thing to do is forced equity dilution of a bank experiencing a run. That is, the Fed buys senior equity in the bank and the injection pays for short term losses.
The equity injection guarantee alone would likely be enough to stop the contagion and calm markets.