Most small banks already have arrangements with bigger banks to cover them in situations like these (usually in the form of a bank-to-bank loan). For example, I know that First Republic has this arrangement with JP Morgan Chase.

What killed SVB was their mismanagement. Per the CFPI, SVB was insolvent at the time it was taken over, meaning they actually did not have sufficient assets on hand to cover their liabilities, even if their assets had been held to term.