The mafia do not know about the alien-invasion-friendly conspirators from The X-Files. That's sorta how things are here.
The Family has influence, and sure, their powerful members in Crisillyir know there are so-called 'reformers' in the Clergy trying to spread a more scientific message that lessens the power of the church hierarchy. But they've never heard the name Obscurati, and they certainly have no idea there's a multinational conspiracy that links weakening the church to opening an ancient seal to other dimensions.
Likewise, the Ob doesn't overreach. They don't need to control everything, just enough to get what they want done without attracting undue attention. So if they want some lighthouses set up, or want access to some ancient ruins, or want the aid of skilled spellcasters in the Clergy, they make contacts within the organization by starting with some basic philosophical common ground.
Like with Ottavia, they said, "Hey, don't you want common people to prosper more, and for the hierarchs of the church not to keep so much power for themselves? We're working toward that end. Here, we'll show you." And they do a variety of small things to advance that agenda, earn the trust of Ottavia and others like her, and link them together into a cell that is focused on making incremental improvements in the Clergy. And if one of those improvements happens to involve removing a church hierarch who's in the Ob's way (even if that requires manufacturing a bit of evidence), well, so be it.
The Ob tricks people into doing things that benefit the conspiracy by couching them in the guise of promoting whatever those people value. Macbannin, for instance, thought he was helping the Risuri military be ready to fight fey titans. Caius Bergeron thought he was investigating ancient portals in order to help the Danoran military research effort. Only those who show themselves to be amenable to the conspiracy's great goals are even told that there are other cells, and only the most trusted get info about the full plot.
As for the Family, there are probably a few low-level Ob spies keeping tabs on what the crime syndicate is up to, to make sure they don't cause trouble for any Ob goals. But it'd be like the Cigarette Smoking Man being worried what the mafia are up to. Their interests just don't overlap very often.
Cippiano, for certain, is his own man. He's trying to use crime and extortion and such to gain power in Flint. He's never heard of the Ob, and if he finds out about them he'd never think to get involved.
As for the long-term involvement of the Family, well, they're not going to show up in Adventure Six. Cippiano has influence in Flint, and the rest of the Family has power in Crisillyir (and a little in Drakr), so we'll next be involving them in Adventure Seven.
They're really not intended to be a 'big agenda' element of the campaign. They matter mostly as a heroic tier factor over the first 5 adventures; I wanted a few easy examples of, "Hey, you might want to consider working with your enemies against a greater foe," since Lorcan Kell's guild is a more obvious threat. Come paragon tier and onward, the Family will mostly just be an 'in' to connect with the Clergy, since they have more broad power.
And the Clergy . . . well, in adv. 7 you can possibly get aid from a priest to help you infiltrate the Ob, and in adv. 8 you'll learn a lot about their history and current conflicts as you're following the memories of Kasvarina. Then in adv. 10 (a bit of a way's off, but I have a general outline) I'm thinking you'll have an opportunity to enlist the Family and Clergy to help battle the Ob.
