Banks are required to maintain capital reserves (distinct from the federal reserve rate) sufficient to weather most financial upsets. And the fact that the federal reserve moves to compensate customers in the (extremely rare) case of bank insolvency is an incredibly good thing, probably among the single greatest stabilizing factor in consumer banking in the US.
It's not actually the fed who does it. It's the FDIC, which is a different entity. The FDIC's money comes from... banks. Technically the FDIC is backed by the US Government and has a credit line with the US Treasury, but they seem to get along fine with the money they collect from banks.
Yes, that's what they're supposed to do. Each of the individual bailouts in that list tells a pretty interesting story: in each case, the FDIC only had to pay out a small fraction of the bank's total assets, indicating that the reserve system functioned as intended.
WaMu, the biggest on the list, didn't require a single cent from the FDIC fund! But again, it would be okay if it did, because that's what it's there for.
