Also, I very much doubt EXALT would be made of human supremacists. That kind of people wouldn't run a subversion campaign against the primary planetary human defense force in the middle of an alien invasion. They're pro-alien, collaborationists, clear as day.
They were only doing what Earth's governments did when they realised they were up against a far superior and near-unbeatable enemy. Cut your losses and secure your position as well as you can for the changed world.
By the events of XCOM2, it's not unreasonable that EXALT have seen and do not like the changes happening to the world, and have the resources to do something about it.
But it would require enemy infighting and I'm pretty sure that's not a thing. So it's unlikely to happen.
So, XCOM recruits are the descendants of EXALT.
As you said, traditional governments would have complied & tried to keep their power.
Their troopers would have, for the most part, stayed their troopers & become ADVENT.
EXALT was a terrorist organization pursuing its own interests, which happened to align with collaborating while the earth govs didn't.
If the earth govs surrendered immediately, the aliens would have no use for EXALT, and for whatever reason their interests are now set against ADVENT and the new world order.
Bradford has some strange bedfellows.