An organization with over $250,000 in cash is not an outlandish amount. Employers (obviously), but also municipalities, schools, churches and heck even grocery stores can exceed that limit. While it is reasonable to expect some individuals to have some sense and monitoring of the financial well being of the organizations they are directly affiliated with it is extremely unreasonable to believe that those same individuals are going to be aware of the balance sheet risks of the transitive banking partners of those organizations. People expect that money in the bank today will be there tomorrow.
Who wants to live in a world where everyone is keeping tabs on which organizations are banking where? It's a tremendous waste of time.
People don't want bank failures to be a thing, and if/when they do happen they want the damage limited to the senior leadership and investors of the bank.
If you were responsible for managing over 250,000$ of cash what is an acceptable Treasury operations strategy? Put it in a TBTF bank (still socializing losses). Split it into multiple banking partners - that creates operational risk in addition to extra complexity and overhead.