The point of FDIC isn't to defend against robbers. It's to enable you to safely store money. You cannot do this with bitcoin. You can do this with USD.
Most people don't walk around with their life savings on them. That is exactly what you're doing by storing your life savings in your brainwallet. Hence, the need for banks.
Bitcoin mixers also make it an attractive avenue for robbers. No one can demand you wire them money without getting caught. But they can demand you give them your passphrase and then make your money disappear in an untraceable way.
