Yeah, this is one of the points that the modern Fallout games really don't get across, especially since they leaped whole hog into the "The Vaults were never meant to save anyone" reveal from Fallout 2

The Vaults were a mix of various social and scientific experiments, ultimately designed to help the Enclave establish a working society which would then make an grand exit from planet Earth and leave all the undesirables (anyone who wasn't a member of the Enclave) behind to die

Would be kind of hard to do this if every single Vault were a deathtrap

Most Vaults were simply controlled populations, never really meant to do anything aside from staying alive and maintaining their population

Some of these Vaults opened earlier than others, and some were never meant to be opened at all (at least by the inhabitants)

But every single one of the completed 100 and change Vaults held thousands (or was supposed to, the shift to 3D made the abstractions harder)

And those are just the known public Vaults, as Vault-Tec was also in the business of making smaller private shelters for both members of the US government and private businesses

The Survivalist's journal entries in Honest Hearts describe how after the bombs there was black radioactive rain, followed by a two month period of such high radioactivity he couldn't go outside, followed by glowing green snow. And all this in a national park away from anywhere particularly hard hit (he counted 7 nukes on/around SLC). Also noted that the Army said fallout should fade out within 2-4 weeks, so months on end of lethal radiation was something unexpected.

Later entries discuss cannibals wearing Vault jumpsuits.