If there is some way to incorporate these factions into an overall scheme to limit royal authority, I say, implement them. Even though they shouldn't act in any parliamentary-type way with regular faction meetings and votes, perhaps a more dynamic system could arise in which the nobles collectively present demands to the king in exchange for their continued loyalty. The Glory faction could push for a declaration of war against an enemy that would not be convenient for the King, or the Faith faction could demand the expulsion of the Jews, even if the King were more tolerant of them. The King should then have to risk the ire of those vassals by ignoring their petitions (and consequently their opinion, taxes, and levies) or give in, but at the cost of undermining his own authority.
Of course, this wasn't very fleshed out or well-thought out, but it is, in my mind, the best way to both present a challenge to the player as well as make gameplay more immersive.
This isn't really the direction we're currently headed in. We don't want to treat them as proper factions with demands and things, but rather as loose affiliations of people with similar interests. Their usefulness is that we can tie your actions as ruler to diverse opinion effects rather than just slapping a -10 to vassal opinion on everything and leaving it at that. They won't demand that you go to war (which would probably be a nightmare to actually implement in any way beyond just being upset if peace lasts too long) or expel the Jews, but they might very well approve of you doing those things. Or dislike those very same things. And with each group having a different list of things they like and dislike, it will hopefully be impossible to make them all happy, which could easily lead to proper factions demanding things and starting wars.
That said, we are talking about various supporting mechanics. If we go this route, we'd definitely be hijacking things like the "burghers asking for city taxes to be lowered" from vanilla and probably adding other similar mechanics.
One issue I can see with making these factions so minimal is that it doesn't seem to allow the player as a vassal to have much influence over the politics of the realm. I appreciate that the player shouldn't have too much power when playing as a vassal but it seems like there should be some way to influence the way the faction acts, I think caocao268's suggestion that the factions should be able to make some demands for their obedience seems sensible but in that case a player vassal should be able to influence whether ultimatums are sent and how they are sent. Have you thought of any way to allow player vassals more interaction with these new factions?
These wouldn't replace other factions, so a vassal would still have all the same options they had before. As I mentioned above, these wouldn't be proper factions in the sense that they're getting together and making decisions. It's more of a way of tracking opinions so a realm isn't quite so homogenous. Player vassals would presumably be able to pick whichever faction best represents their interests, and if we do add any supporting mechanics, they'd get to interact through those as well.