The $42B that depositors pulled out of SVB has to go somewhere. FRB is the next-best-choice for many startups; they are also startup-friendly, have plenty of branches in Silicon Valley, make it pretty easy to open an account, many of these companies may already have accounts or banking relationships with them, etc.
This is one way in which the current crisis is self-limiting. It was brought about because SVB took lots of deposits when rates were low and invested them when bond prices were high, and now they don't have liquidity without taking a loss. But if banks take lots of deposits now and invest them when bond prices are low, their average cost goes down and it's much easier for them to satisfy customer withdrawals without taking large losses.
It becomes a crisis again if all the banks start failing at once, because then the likely consumer behavior is a flight to hard cash or crypto rather than another bank.
