Incorrect-ish. 12 U.S.C. § 1821(f)(1) states that payment shall be made by the FDIC "as soon as possible," but as my sister-the-lawyer has always been at pains to point out to me, the most up-to-date specific language is from 2016's 81 Federal Register 87734, 87735, section B, which explicitly states that they "[do] not specify a time period for paying insured depositors." In other words, if it's a tiny bank failure you might get lucky and get made whole in a few business days, but if it's something larger, they can choose to meet their regulatory obligations over the course of decades - which is very much real risk.
I mean, I get it, but that would destroy any credibility of the insurance, and would undermine the whole point of the system.
I trust the FDIC to be far more johnny on the spot to resolve a liquidity issue than I do any private entity