It is and it isn't the depositor's responsibility. It's totally expected for a large company to take a long, hard look at their bank. When I was an undergrad in Econ and in Accounting, the issue of insured account limits was literally in the text books. In the accounting/finance/economics area it's already well understood that a CFO (or their office) is responsible for vetting the bank. Moreover, it's the CFO's job to make sure the bank has adequate controls because (unlike consumers) commercial deposits are not indemnified when fraud occurs. If a company's payroll is stolen through fraud, for example, it's likely not recoverable. (Unless the bank failed to follow their procedures or the procedures set forth by the depositor, and you may still sit in court for years.) If your personal Visa debit card gets compromised, you aren't liable for the fraudulent purchases and the processor eats the cost. The health of the bank is less opaque to mid to large businesses, which have access to tools like Lexis/Nexis, Bloombergs, detailed ratings, and let's not forget personal networks. Most large companies also have to run big choices, like what banks to use, by their boards.

. Pending litigation
. Counter party risks (e.g. do they have exposure to a problem bank)
. Financial statements (usually the first stop and contains a lot of information)
. Credit Default Swap rates (what does it cost to insure the debt issued by the bank)
. What rules does the bank operate under (domestic, foreign, state, federal, etc.)
. General reputation in the industry (e.g. go to $CS to launder your cocaine money)

That being said, it's probably beyond a small business to really evaluate their bank risk. And while 250k may have been adequate for the 2010's, it may no longer be sufficient to cover many small businesses (and by small I mean the back office is a handful of people). Should it be 500k? 1 million? I don't know. What I suspect is that making it unlimited, it means that a bank that's hemorrhaging depositors could offer unsustainably higher rates. Less risk-sensitive CFOs might decide that parking 10 million in reserves might be a good deal since it's as safe as an officially insured deposit. It's much more liquid than holding a 90-day CD or 3 month T-bill to maturity. Suddenly, the bank is flush with deposits, but is still going under. This would be kind of like the 1980's S&L crisis. That's why I'm all for raising the cap, but with clear limits and adjustments to the fees charged to member banks for insurance. If we need to expand the FDIC insurance fund by 3x to raise the cap, that's fine. But raising the cap, for an extended period, without adjusting the rules or the price for the insurance could lead to unintended consequences.

Should a VC funded company of 10 people do what GM does when evaluating a supplier or customer? Probably not. What about when it gets to 100 people? At that point I would expect there to be a competent CFO. What about the VC? (I'm just going to ignore all the tweets from the All-In community that showed a profound lack of understanding of banking, confusing a modern bank Gringot's.) Should the VC, as part of their advisory role, maybe recommend a good part time CFO or cash manager? Did regulators screw up SVB? Possibly, there were a lot of issues found when they transitioned to a new regulatory team year or two ago (or so I read). But regulators are not bank managers. And if a bank can show their risk controls are adequate, and those risk controls are being followed, it does not mean they're making good investments. (It's arguable that both lack of controls and following controls were an issue for SVB - as far as I've read).