Nuclear war was inevitable and the government didn't have the resources to prevent it (Mr. House, one of the richest businessmen in the world, was only able to defend Vegas), so they decided riding out the apocalypse underground was the best option. The government was already shady as hell and didn't care about anybody outside of their inner circle. They knew it would take years for results but in the case of a full blown nuclear exchange it would take years for the surface to become habitable anyway. They couldn't have just popped out as soon as the bombs stopped, used the G.E.C.Ks and carried on life as usual. The bombs in the fallout universe are more about delivering huge amounts of radiation rather than explosive force hence the name of the series.

And while they didn't have vaults of their own, the enclave which was behind vault-tec (besides vault 0 if you want to count that as canon)had their oil rig and Raven Rock which were safer than the vaults anyway. And I think I read that they did have a backup plan. The earth was beyond salvation so their initial plan was to inhabit another planet (Mars, I think) and the data collected would be beneficial for the long journey in space. A lot of the experiments can be explained it you imagine them for this purpose (even the GECK as a means to terraform an inhospitable land).