Agreed, the player character being in many ways so special really does hurt the influence system and obviously is immersive breaking. Off the top of my head, other lords will very rarely propose votings for adding new/removing existing policies, won't propose a fief to be given to other clan, as you noted won't enlist the player to their army(but when getting enlisted themselves will never object and always join) and so on.
These could be improved in ways the game already does in some places - lords decide not to join your army due to clan relations or their personal traits, player and their clan parties also gets drafted into an army, if refused lose relations, influence etc. If a clan has low relations with another or is entirely unpopular in the faction, they might propose to give away their fief to someone else. Policies could also be proposed more often, with a clan's culture, leader's traits and overall things like that dictating which policy will be brought up for voting.
Maybe more ways of getting influence, especially ones that AI clans will be able to use? Clans could set a priority for their parties to acquire influence if they were low on it for enough time. And of course more exciting things to spend influence on would be nice to have, but I'm probably going too far into the "ideas for mods" area already. Some of my ideas would be clan upgrades(or faction ones), settlement upgrades, unique gear to act as endgame influence-sink, other lords also having companions in their parties and faction leaders could create a clan out of a companion, just like the player can
