The underlying issue is that even uninsured risks in banks must de-facto be largely covered by some central authority (whether FDIC or otherwise) otherwise the whole system is at serious risk of pretty catastrophic failure. The interaction of the lack of (formal) insurance for large deposits and the fluidity of those deposits make bank runs at the first sign of serious risk inevitable.
How important that problem is - I'm not informed enough to have an opinion on. But that problem didn't start in 2008, that's for sure. I am curious as to what actual subject matter experts think ideal solutions would be, and what haphazard lobbyist inspired patch-job we'll actually get.