Who needs the FDIC? People are certainly free to purchase insurance on their Bitcoin and more (better) options will certainly be available as time goes on. I doubt the FDIC would want to be bothered, simply because the FDIC is an agency whose mission is to protect the American dollar and the banking system.
The whole point of Bitcoin is to decentralize money. The problems that come up should probably be solved in a decentralized way. I don’t think anyone who understands Bitcoin would suggest that you should keep your money on a remote server run by someone else.