That didn't prevent multiple runs against Washington Mutual in 2008, and it won't prevent runs now. People are worried that they won't be able to get to their money even if it is insured.
Plus, even if the "typical" consumers don't freak out, businesses might. There are a lot of businesses with accounts over the FDIC limit. Only about 60% of bank deposits in the US are insured.
If I thought the US government was going to fail I'd simply try to be the first to convert my dollars to a dense and valuable commodity, like gold, and then find another government to live under. It's not like people pulling out dollars have to keep their wealth in dollars.
How would they even do that? Bitcoin handles ~350k transactions per day. VTI (Vanguard's Total Stock Index), a large size fund is on average traded 3,670k per day. That's one fund. There is no reasonable way for small customers to pull out of index funds to bitcoin unless there is a fundamental change to how bitcoin operates.
Small customers get BTC from an exchange, which keeps their accounting internally. No transaction volume limits.
Later the small customer may move that BTC to self-custody, which could be an on-chain transaction or LN or other side chain with more transaction capacity.
Not typical consumers, right? Typical bank consumers have < $250k in their account, and thus there's no reason for them to cause a run.