There is no such thing as a civilian organization prepared to jump into immediate action to handle a large scale disaster of any kind, period. Even the military needs some time to operate on a national disaster scale operation and they can launch limited operations within hours.
To expect that somehow if the gov't had been smaller and less money being wasted would allow the federal gov't to supercede local and state gov't operational plans and somehow rescue 100,000 people hours after a disaster is fairly ignorant of reality. It has nothing to do with gov't size or layers - it has to do with local, regional, and state authority coordinating with Federal agencies to augment a plan that needs greater assistance levels than can be provided at the local and state levels.
BTW, if you are expecting any gov't on the face of this planet to come to your immediate rescue in any disaster, you are in for more bad news. Local churches and church supported organizations always accomplish much of the relief efforts in these kinds of situations and are never recognized for their efforts.