Isn't this exactly the problem the FDIC solves? Nobody with less than $250K need worry about who they bank with if they see the FDIC logo.
The problem of regulators being unable to proactively stop a crisis (and then using that as part of an agenda to expand their powers further...before they fail again, and then say they could have stopped it with more power) hasn't gone away though.
Btw, if you doubt this, I have already seen stories briefed into the media out of regulators saying that this happened because Congress passed a bill lowering the threshold for stress tests...ignoring the fact that SVIB was still being overseen by three distinct regulators, was still doing stress tests, and the problem concerned extremely liquid financial instruments that were reported in full in their totally public reporting.
I believe the FDIC has come through 100% of the time since it came into existence in the 1930s. Here's what they say: "Since the start of FDIC insurance on January 1, 1934, no depositor has lost a penny of insured funds as a result of a failure." I believe that's true.