The FDIC exists for exactly this reason. As we saw in the case of Silvergate, SVB, and Signature, the system works well. If you were an insured depositor, you knew shortly after the FDIC put the bank into receivership that you’d have access to up to at least $250k on Monday morning.
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Since the sweep network distributes cash across the network, your deposits up to an aggregate level of $50m are insured by the FDIC.
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In the example of a single bank failing, $250k of the initial $50m would still be yours as a customer, since you were fully insured at the time of failure. From then until that failed bank is replaced in the network, your aggregate level of insurance would fall to $49.75m.

An FDIC claim would be filed on your behalf for up to $250k in the account at the bank that failed. Typically, such a claim is settled within two business days or less. All other funds remain fully available.