After the big Rodeo-Chediski wildfire here some years ago, the biggest time consumer, and most profitable for the homeowner, recovery activity was sifting the ashes of the homes. We sifted ashes where the bedrooms or den had been, or other rooms identified by owners as places where valuables were kept. We recovered lots of jewelery and loose gems, rare coins, etc. that would otherwise have been lost to the front-loader scooping up debris for removal. Other than that, nearly everything was a total loss. Rare exceptions were inspiring, like the people who's family records had been in a chest of drawers in the basement and a broken water pipe had hosed it all through the fire, preserving their priceless old family records, photos, etc. Generaly, not much is left after a major fire.
Tornados and earthquakes make a mess, but tend to leave more salvagable stuff, making staying on site more worthwhile. Tornados, as we had in Colorado, may put lots of your stuff in a neighbor's yard, but people are usually very helpful in such circumstances. It's refreshing to see how people come together and help each other in such trying times.
After major floods, which we had often in Missouri, the property is pretty much totally lost, and staying on site is often difficult due to health issues, and lack of services and solid ground on which to set up camp. Some property may be recovered, usually small items that need to be cleaned.
In any event, there's lots of work to be done, and staying on site helps with planning, reduces travel time, and ensures someone is there to direct recovery efforts of volunteer helpers. If possible, I think it's better to stay on site.