First, it needs to meet more often. every 15 turns or so would not be out of whack.
I agree with this based on how the UP is now. It already takes so long to meet enough races to even get the UP started, depending on the galaxy size and player number. That by the time things really getting started in the UP it already feels like you have waited forever. If their are more interactive UP choices, even something similar to my idea of Projects. The longer wait times between UP votes might be all right as instead of getting a random vote and then nothing for ages, you could be spending a few turns focusing on doing something UP related which would make the UP feel like things are happening more often even if they are not. Not to mention if there are less turns between UP votes or Project Events then the mid to late game will feel less "samey" as there will be constant new choices and self made goals that your actively achieving. As long as it dosnt feel like ALL your doing is UP stuff all the time of course.
The note that the UP meets in X turns is basically nothing... we the player cannot do anything with it. Once a vote is called we have no more influence than our own Civ influence.
As has been been suggested before but I always liked the idea that we could buy votes either with credits or trade from other civs. Influence should always be a big part for the UP giving you more sway. But if the UP is nothing but an influence game with no consideration for income/bribes, Espionage, etc. Then its just something that happens every few turns that your not really participating in. As you say its just "there".
One thing I think is overlooked with the UP is that since only 1 race is the chair and has the say on what is proposed then every other civ really feels like its just watching one player having fun screwing around with the UP. Civ 5 had the good idea of allowing the top 3 influential civs being able to propose things as well that are then voted on. Something similar to this is even more important in a game like Gal Civ 3 where unlike Civ 5 where you might just have a handful of Civs, you can have upto 100 other civs meaning the bigger the game the less relevant an interactive the UP is. I feel their needs to be some sort of balance based on player count.