"Influence" should ideally only matter in comparison to what other Lords have, though right now it works just like a currency, irrespective of what other Lords have/do not have. If I wanted to "force" a very influential vassal to stay with my army, that ought to cost (a lot) more than keeping someone who just joined. But yes, CotC is way too good, it accounts for roughly 100 out of the 120 or so influence I receive (passively) in my current save.
However one policy generating large amounts would be less of a problem if things weren't capped at specific influence costs - I may have ~30K influence, but I have several other Clans who have 15-20K, rendering the whole system pretty meaningless - it's not like those other clans can exercise more than ~300 Influence's worth at a time anyway.
It reminds me of Stellaris: Federations Galactic Council mechanics. There you have policies, which empires can enact and they do it by comparing sum of diplomatic power of empires voting for and against said policy. Diplomatic power is sum of their technology/military/economy x previously enacted multiplier policies (for example +20% power from military or -20% from economy).
Additionally you can buy and sell "favors" for any other resource or pact. One favor is worth 1/10 of empire's diplomatic power, so, with help of good economy and diplomatic strategy and planning, you can become the senate.
I think it'd be nice to see voting as some kind of diplomatic battles rather than just spending 50 influence as level 2 clan and having not less control than level 6 clan, which spends the same amount of it.
What about coherency then? It could be nice decision making there: You have a choice, whether you want to save your influence to have more power over resolutions or to spend it and get some good ol' army for few more days.
Then, policies would need to be less universal and be punishable for one and rewarding for another lord. In turn lords would have to be 'conscius' of their situation and potentially best moves for themselves. It'd make internal politics much more enjoyable and competitious, at least for me.
Of course I don't say 'let's rip off Stellaris', but I think there, and in other Paradox grand strategies, devs could find some good inspirations.
EDIT:
Maybe Clans would use few hundred influences to boost relationship with some other clans within the faction, if they happen to have more than 5k influence. This could create more dynamic gameplay specially after Rebellion feature is added to the game, that, independent city-state clans could invite other clans to their factions easier with good relationship, and there would not be influence inflation. I currently use a mod that caps the influence at 5k so no clan can get more than 5k influence, but i would prefer clans to use excess influence to get relationship with other clans to deal with influence inflation rather than using a mod.
In system described above, I'd see it as spending favors for relations, so somebody else could just use your diplomatic power to have more control over voting and then you'd have some good relations with them.
I think that influence is mostly internal politics mechanics at the moment, but maybe it'd be better if it had wider approach.