I'm not really sure what their plan was in 111, besides the cryonics.
As far as I am aware, all the info in the vault just says that the test subjects were to be monitored, possibly indefinitely.
I don't think there was anything specific about the Lone Wanderer and his family that meant Vault Tec really needed them, other than the fact that they lived very close to the Vault.
Maybe having a baby/child was important, but probably only in order to test out the cryonics. There weren't any other children in the vault, so that seems possible to me.
As for rushing to get them into the Vault, I'm guessing that because Vault-Tec was connected to the military, they had early warning of the bombs, and were rushing to complete any remaining files/checks needed in order for their test subjects to get in to the vault as easily as possible.
That said, the Vault-Tec rep only wrote down your results on a piece of paper, and then seemingly carried on going to different houses.
And when the bomb falls, he's just trying to get into the vault, so he's not exactly handing anything over.
I guess it could be assumed that we definitely were on the list anyway, and he was just doing some last-minute checks, or at least informing us we had a place.