In a sense you can say the Enclave had a coup, and the Great War was the catalyst.
Not really, you could say that there was a global nuclear war and that these so-called "Shadow Governmenters" had legitimate positions in the United States which meant that they were legally in-charge - you know - how continuity of government works in real life? Shadow Government my [censored], a lingering - never utilitsed - plot point
from a man coming off an X-Files binge; you know how big the Shadow Government was in the X-Files? Less than 2-dozen old white-guys, mostly from the State Department, whom were assisting the soon-to-be alien invaders in-creating an Alien-Human hybrid slave race so that they personally could survive.
Now we have this, "Shadow Government" plot without the whole " We're abandonning the planet to go to space" point and people keep coming up with "They wanted to rule the U.S. Government proper, which as a pretty pathetic Shadow Government they apparently cannot do". Manipulate the entire Vault Project? Easy and pie. Derp, us high ranking officials apparently immoral enough to do anything not get voted into the Oval Office - that's a hard one! Mind you, not as hard as planning your evil long-term survival plan aboard an Oil Rig with only yourselves and your families.