The whole illuminati angle, IMO, is kind of dumb. It is like the cartoonishly evil badguys of Bond.
The enclave, was supposed to be a SMALL cabal of corrupt power brokers who might not have had all that much power pre-war. As things got worse, like in the Fallout verse, non-enclave personnel died with the rest or the population.
They got more power as those who could challenge them died but it was certainly no monolithic secret entity. The bigger you are, the easier to be found out.
I always imagined the enclave like the sith, post darth bane(?), and their rule of two or the NSDAP. Lets keep NWO, Tri laterals and FEMA deathcamps to Deus Ex.
And second Dr.s idea on the oil rig. It was a last ditch emergency measure. A bunch of enclave personnel retrofitted an oil rig and waited out the shit storm on it. Hell, they probably were so small and secretive that many didn't even know they existed or, atleast, were not considered much of a threat taking account of world wide events at the time.
When the REAL, COG was wiped out by the nuclear exchange, they formed their own. This explains how a high value target like the president survived, the guy certainly wasn't important pre-war, atleast to the chinese.