1) FDIC insurance only applies in situations where the bank doesn't have the assets to make depositors whole. SVB has a ton of assets; most sources I've found asserting 100% deposit coverage, just not liquid. Even if the FDIC takes over (which isn't even likely) (edit: this aged well), the insurance element is irrelevant; its about operations and finding funding to drive liquidity.
2) When startups close a round of funding, they don't just get a check for $20M and throw it in their SVB account. Funding rounds are an agreement between the VC firm and the startup for that money, which transitively represents an agreement between the VC firm and its partners, and the money is generally delivered "just in time", not all at once. When the startup needs an infusion, they go to the VC, who then goes to their own bank accounts or their partners, who then go to their own bank accounts, and wire transfers happen. SVB is only one player here; yeah, its absolutely true that many startups (maybe most) directly use SVB, but its less common the further up the chain you move as the money gets more and more boring (when you hear "partner" think "old boring local business magnate who has banked with JP Morgan for 50 years"). And more-over short of systemic bankruptcy the VCs are still on the hook for that $20M.
The risk that SVB, the financial industry, and regulators are worried about right now is short-term liquidity. Startups may have $xxx,000 in their SVB account which they use to make payroll every two weeks and pay vendors and such, which is separate from the $xxM on contract with the VC. If SVB can't meet outflow demands, the people staying and trying to make payroll are going to get caught up with the panic'ed people trying to pull all their money out, and short-term liabilities like payroll are at risk. That's part of of the reason why some VCs are pushing their startups to pull money out; its not about "oh my god we're going to lose all our money", its because they don't want to get caught in the herd and be forced to pull money from other sources which are also less liquid, like long-term investments or going to their partners. Put another way, SVB's liquidity issues could spiral to cause liquidity issues further down the chain; and no one in the industry wants that to happen.
But, its a macroeconomic prisoners dilemma. And, to be frank, and I mean this absolutely genuinely and sincerely; most VCs are just rich idiots. Lets be real, the industry is proud of the fact that if one bet in fifty pays out 100x it'll make up for 49 bad bets, yet we treat them as paragons of investing genius? Their biggest motivation is to avoid embarrassment among their drinking buddies (read: investing partners) during the next trip to Jackson.