Eh, I usually go by Mr Enclave in the Fallout Artwork section, wish I could change it to that now to be honest.
Actually, as "The Enclave" I know that the Oil Rig was the only major facility occupied by members of the United States government, President Eden mentions that "decades after the War" such facilities evacuated to the West Coast when your discussing things with him. Besides, how many people do you think that there are in these bunkers, they aren't like Vaults designed for centuries-term survival.
And Chicago is described as merely an "outpost" which means it could be anything.
EDIT: Not only that but if some hypothetical facility managed to continue on it's own in the 2270's they wouldn't be the Enclave would they? Just some group of pre-war survivors. They'd have nothing to do with their culture or mission. Considering that the mission in Fallout 2 was to kill everyone outside of the Oil Rig/Navarro then that would also include them.
I want the Enclave to die-off because I'm sick of Bethesda ruining them. F3 and Broken Steel were disgusting.