In the case of SVB, it seems that their assets will cover their deposits, and FDIC has an insurance fund that gives them the liquid capital to cover deposits immediately while waiting for assets to sell. So no cost to taxpayers - not even in the form of “higher deposit insurance costs to banks being passed on to bank customers, who are taxpayers”.
(It’s possible to play semantic games until we formulate a picture that does show taxpayers will pay, e.g. “the FDIC’s deposit insurance fund is made up of payments made by banks, and banks would have passed on the cost of those payments to customers in the form of not offering as much interest on deposits as they otherwise would have offered, so the funds used to make the bridge are a bailout the taxpayer has already paid for”. Money is infinitely fungible, you can always tell a story where taxpayers paid the bill. But we said we weren’t going to play semantic games.)