Even though I liked the basic idea, I kept away from these topics since a first look at them felt like it's quite complex.
Well, now that I have read this one and some more, I can tell that it actually is
I agree with half of the propositions even though they contradict each other, I don't understand some of the discussions (with a good intersection with the former affirmation), I don't know where some of you are going and apparently some of you don't want to go anywhere.
The only thing I can assert right now is that the Species-Empire Relations is indeed needed, for the reasons Vezzra exposed
Oberlus' proposition to get them from an average of Planet-Empires Relations may be a good path of simplification, but I feel like they would still miss something though I can't point what exactly.
Anyway, one thing we can all agree it that their effects should not be boring.
Which, in my opinion, means that they should not be linear.
We should have situations where species suddenly begin to demand things from the Empire (obviously contradictory demands to make it fun) and let the player trying to cope with it while trying to not entirely derail his own strategy.
Obviously not completely at random lest FreeOrion becomes just a game of whack-a-mole, but still without the predictability of a spreadsheet.
And obviously, Influence should be important to manage these competing demands and the detrimental effects of not satisfying them, but it shouldn't be a simple and boring "pay this much to have that" (i.e. if your Furthest are unhappy about the war that's coming closer to their home, you can't just pay X influence and be set).
At this stage of my reflection, I think the best is to use two "currencies", Influence and Time, and to use them differently.
What I mean is that Influence would be able to buy you time, but only that : if your Gysaches are demanding you to stop the war, paying Influence will allow you to get a few more turns of stability to achieve strategic objectives before ending the war, but only that¹.
And once you have upset one of your species, only time can be the healer: each turn in which you meet this specie's requests and have policies that makes her happy, you get a rising chance of having it revert to a better appreciation of you.
¹ So I guess the best way to achieve that would be rising influence costs as long as a demand is not met.