Where did all the humans come from?

The Fallout TV show makes it out to be that rich people went into vaults while surface dwellers suffered surviving radioactive hellfire, but that makes no sense?

The nukes killed everyone immediately. We can see skeletons everywhere, so no one would survive anyways.

And if almost every vault was a fucked up experiment, then not many would survive that either.

So where tf are all these people coming from? You can't seriously tell me everyone came from vault 76 (that the vault for fallout 76 right? Or am I mistaken?)

Then there's New Vegas I guess. And rural communities.