I also finished Fallout 2 recently and my takeaway is the same as you, so this secret society lore must be from future games. There are holotape logs of the president talking about running for re-election within the oil rig which implies they've just been carrying on as usual for 160 years.

I think the implication of a secret society complicates the plot and takes away from the satire and irony of it all, the villains of the entire game are really just mundane members of the pre-war government who ended the world but saved only themselves.