Have I got a story for you.
What you described normally happens.... Unless the bank fails so hard that no buyer for it is found, and then the FDIC insurance kicks in.
I know, because it happened to me back in 2012. :-) My bank failed, and they were in such bad shape that nobody wanted to buy it! I got an email on a Saturday morning stating that my bank had failed. Here's a copy of said email: https://imgur.com/a/HR42BxZ
The following Thursday, the check from the FDIC for the contents of my bank account arrived in the mail.