The FDIC doesn't deal much with bank customers on an individual level. They also don't take over banks persay - they help liquidate them and sell the business to another bank that can handle the accounts. They are very, very good at what they do. There are occasions when certain government bodies can be small and have a very confined focus, and all their employees are highly experienced experts.
Occasionally, if a bank is very bad and no other bank will buy it, they will send checks to the customers.
It really bothers me when people complain about the FDIC and claim the free market could do better. The banks would scream bloody murder if the FDIC was instantly dissolved - even if it's assets were split among banks. If the FDIC goes away, then consumer deposits are not safe, customers won't make deposits, banks won't have money to loan, and much of the economy stops. A private for-profit company will not come in and save the day during a bad economy - for-profit companies don't do that. But we can imagine a scenario where there is a company that does offer private insurance for bank deposits. An insurance company would be the obvious choice, as this is their business anyway and they are usually very conservative with money - and many provide mortgage insurance, so deposit insurance would not be a big stretch. And in 2007, the largest, wealthiest, and most capable insurance company on Earth was... AIG.
So, overall, I think the FDIC has done a pretty good job.