Yes, but depositors are generally not expected to be sophisticated investors in the context of bank liquidity abd solvency. You don't get a prospectus of the bank itself when you consider opening an account.
But someone that has over 250K in a bank can be expected to understand that FDIC insurance is (or better was, effective now it is unlimited) 250K, right?
First, no. My wife earns more than me, but phrases like "FDIC insurance" make her eyes glaze. She'd ask her parents, "What do you do?" They'd answer, "Keep cash under your mattress."
Second, what does that really mean? How to evaluate the quality of your bank isn't typically discussed by personal finance teachers, beyond looking at the interest rates and fees.
The prospectus is easily available, for all the good it would do.
In any case, you don't need to be a sophisticated investor as a depositor. Ordinary market participants manage 'flights to quality' just fine, even if they are not sophisticated investors. See https://en.wikipedia.org/wiki/Flight-to-quality
If you want to have a stable financial system, you shouldn't suppress the incentives for people, including depositors, to look for safety. Just the opposite, you should have them sensitive so that they make moves (on the margin) long before danger is serious. Have people move their deposits _before_ it's too late.
More seriously, you don't need to do extra due diligence. Just buy insurance for your funds above $250K. Or use standard treasury methods like using institutional insured liquid deposits or sweep accounts.
Liability insurance is required for automobile drivers, because of the externalities. Perhaps bank run insurance should be required for depositors, because of the externalities. That'd force depositors to internalize the externality of banking with a fragile bank.
