It’s hard to imagine how irresponsible these VCs and startups were.
If we insist that they take on their bank's solvency risk as depositors, the rational decision for most of them will simply to be move to a systemically important bank that the govt will likely never let fail (Citi, BofA..). There's no reason for them to keep their business at a small bank and take on risk they don't need to. Regional banks would then wither, and the banking industry would become even more concentrated with a few too-big-to-fail banks becoming even more powerful. It would be a bad outcome for everyone.
BTW, for anyone wondering which banks are considered to be 'systemically important', here's a convenient link to the list:
https://www.fsb.org/wp-content/uploads/P211122.pdf
And here's a link to the group that decided who goes on that list:
Those people I know who run restaurants basically have zero cash at hand and are constantly repaying some debts.
For the vast, vast majority of startups that's not their business. Storing a few million dollars should not require expert due diligence into a bank's holdings when you're in a developed country, due diligence that these business owners don't even know how to do. It's not irresponsibility, it's focusing on the things a small business should focus on rather than distractions.
How often do you assume that your prime broker will go out of business before your hedge fund does? Unless you’re a big one (Citadel, Millenium), probably “never” (it’s the same with startup companies).