I would say that Vault Zero was indeed a place for people such as high ups in Vault Tec to survive in the event of a Nuclear War. Fallout Tactics says "The Great Vaults were built to house the wealthy, the powerful, the influential and those deemed necessary for their survival. Inside resources and technology were stockpiled. A final defense against the coming holocaust."
Now we could argue over what "The Great Vaults" means. Does it mean all the Vaults or just a select few vaults. But the intro goes on to explain Vault 0 after that statement to say: "The sturdy Vault 0 was to be the nucleus of the vault network housing the greatest leaders, artists and scientists."
Vault 0 was going to be center of the Vaults. It was to direct the rebuilding of society. But we know that the Vaults weren't built to save anyone. The government used Vault Tec to build the Vaults. We know that there must have been scientists and powerful, wealthy and political people who knew the truth about the Vaults. They must have wanted a place to survive. We also know the Government couldn't sell the Vaults to the American people by telling them the truth about the Vaults. We also know the truth was that key people in the government were going to go to Oil Rig to ride out the Great War.
So it makes sense that Vault 0's role was all a lie as well. It's main job is to show the American people that there is a system in place to insure that America will be safe. That the Vaults would be connected with one another. The top minds in America will rebuild America. Another role would be is that the powers that be who know about the Rig could tell unwanted people that "yeah we have a place for you in Vault 0. You are going to be a very important person! "
It is possible that Vault 0 failed by mistake that the plan was always to have two locations for the Powers that be to survive the War. The people on the Rig calling the shots while the people in Vault 0 do their bidding .
But it is possible that Vault 0 was never meant to be an important place. That only second rate people who knew to much were sent. I like to think the money to upgrade the Enclave Oil Rig came from cutting corners while building/upgrading Cheyenne Mountain Complex.
In the game it is made clear there were several cut backs. If what I recall is right, billions of dollars were cut from Vault 0 because the Powers that Be decided there was little chance of a nuclear war. That money for important systems in Vault 0 was diverted for luxury items and entertainment. It would be very easy for money to disappear from Vault 0 and end up for the secret Oil Rig project. We know in Fallout 4 at least one Vault, 114 is rumored to have been an embezzlement scam. If you are going to have a secret offshore Doomsday Bunker, you will need a secret source of money.
Edit: I like to think Vault 0 was designed to fail. A means to ensure that people were indeed dead. Can't risk that they might survive and ruin things.