Urriak Uruk
Gaming is fun, and fun is for everyone
If you give each faction about 2-3 pages pages, that still leaves room for the player rules and details on Sigil (or at least detailed at the level of a book like Van Richten's Guide). Essentially, the Factions could be treated like the Domains in Van Richten's Guide, as the drivers of plot in the City and Outlands. Then the Tour of the Planes Adventure can show how it all works in practice.
Oof no thank you. I'd much prefer each Plane in Planescape gets the Domains treatment, not the factions. I'm looking for a Planescape setting book, and if its that many pages devoted to just factions in Sigil, that's way closer to a Sigil Setting Book. I'd be very frustrated with that.
I largely agree with @Ruin Explorer that if each faction gets 2 pages, that doesn't leave much space left for the rest of Sigil, much less the rest of Planescape. Especially if it has to share space for monsters and an adventure, which a Planescape book probably will have.