If we didn’t call out a particular body as being important to our story, we’re all agreeing it’s not going to be important. In mechanical terms, faction = important.
Now…if the story progresses such that everyone agrees some new group should be considered important (i.e. can be controlled or destroyed), then between Phases I suppose you could negotiate to add a faction to the list. But that’s one of those things where the entire table would have to agree to edit the shared narrative.
And finally, if “include factions” wasn’t chosen and you DO want to include factions later, that’s sort of dirty pool since the Vaylen side gets a disposition advantage for even addressing the issue of factions. I’d probably be asking how the world burning phase missed the mark so completely if that came up. My first guess would be that someone was trying to game the system in a way that wasn’t intended.
In your regime change game (awesome theme BTW), I’d absolutely make all the important groups in play factions during World Burning. One of those factions will be in power at the start of the game, per everyone’s agreement. The Infil Phase intent will probably be “change government” and, if successful, a different faction will be in power in a following Phase. However, they’ll all still be factions and therefore you can control them, shut them down, use Circles to make NPCs within them.
Note that at the top of each Phase, the players will have to re-Assess and re-Take Action even if they had control of the faction in a prior Phase. Therefore, you couldn’t, say, Take Action vs. the Military Junta in the Infiltration, put the Junta in charge at the end of the Infil as your Phase Intent, and then pow! be running the world during Usurp. All factions would be back up in the air.
What you could do is set up the narrative during Infil so that your PCFONs had Affiliations to the Junta, Reputations within the Junta, you had Circled up useful NPCs within the Junta, and had otherwise set things up so the Vaylen side would be hard-pressed to make good use of the Junta themselves during a future Phase. But what’s good for the goose is good for the gander: the Vaylen could also Take Action vs., say, the Junta, hull every one of them, and leave you with a hulled population the PCFONs couldn’t reasonably “control.”