Here's what you get in the current model: When you brainwash, you gain access to the assimilation trait and +25 leech per turn.

Incidentally, the assimilation traits are still very much in the game.  You can see them on your Empire screen.

I too would very much like to have something to pour influence at. In my first game as the Vodyani I had about one gazillion influence, mostly because producing it allowed me to cleanse planets (of inferior life forms) without having to get my hands dirty (I like to think they all commit mass-suicide because of how awesome I am).

In EL, Empire Plans scaled with how many cities you had and you generally wanted to spend as much as you could. Unless you were playing Drakken or Morgawr, you'd spend almost all your influence every empire plan.

In ES2, basically everyone (except UE) gets a huge influence surplus in the lategame. I'm not sure if election action costs scale with how many systems you have right now (they probably do), but they should probably scale even more. Another thing to help with this might be adding influence inflation in the same way it's applied to dust. People get used to your culture after all. Perhaps diplomatic agreements should have scaling costs so that they're not trivial switches to pull in the lategame. In any case, I'm sure they're aware of this stuff and will do some balancing around it before release, but right now it's not that big of a deal. It's a problem, but a pretty small one.

Here's another option: Why is tech associated with how many law slots or senatorial slots you get?  Why not influence? We have a screen for spending manpower (not tech!) on improved infantry and tanks, why not influence to improve your government?

If we want to make sure people need to spend buckets of it, you could require said influence every election.  For example, say you have a Dictatorship with 1 law slot and 1 senatorial slot.  As a dictatorship, you can never increase sentatorial slots, but perhaps you can spend 10,000 influence for +1 law slot that lasts until the next election.  Perhaps as a democracy, you have 3 law slots base and 2 senatorial slots, but you can increase senatorial slots at 1,000 influence per slot, and +1 law slot at 2500 per slot.  This would represent how well each "regime" manages their government, and you could spend said influence immediately following the election, after you see if the results are worth it ("Oh great, I was able to hold onto Science, but hey, industry came in second, that's an interesting result, let's buff my law slot options and pass a bunch of laws for the next 20 turns")

Someone elsewhere on the forums suggested capping Influence the same way you cap Manpower, forcing players to think very hard about how to spend their influence, because even if they produce a ton they still have a hard limit on how much to spend at once. I think if that were combined with making deals Influence/turn rather than a lump sum, you'd get interesting tug of war in politics, so for example when dealing with Minor Factions you would allot an influence budget to them, in an effort to slowly outpace any competitors who are also courting that minor faction. That way you can't just snipe them, though you'd then need more complex and nuanced Minor Faction diplomacy benefits to fit the subtle and slow nature of this system; currently, it's all-or-nothing benefits to fit an all-or-nothing diplomatic mechanic with minor factions.

I do very much like the idea of turning the Senate into a playground for Influence use, though! Make it sort of a very complex, constantly interactive aspect of the game; spend Influence not only enacting laws, but going out of your way to CAUSE the kinds of political events you normally would get randomly. You could even enshrine certain laws as longer-term Programs or Departments, like how no matter which parties are in power, there are some government services that no politician in their right mind dares mess with, like retirement benefits or medical aid programs. Yes, that medical program began as an Ecologist law, and the Industrialist party is in power now, but that doesn't mean it wouldn't be political suicide to try and abolish it. So you could upgrade an existing Law by allocating a permanent budget of Influence and (maybe) Dust, allowing that law to survive multiple administrations as long as the budget keeps rolling on; then a new random event could be the current Party showing hostility to the law, and trying to get rid of it, requiring a lump sum of something or other to fend off their budget cuts.