I appreciate the update. Accurate BGS info appears to be... Somewhat elusive.Almost correct. The two things that would continue to happen are Famine and Outbreak.
FD confirmed a while back that they put in a "Natural increase" in the occurrance of Famine and Outbreak that is (typically) countered fairly easily by player activity... but with no interaction, factions will eventually hit a famine/outbreak state.
This was because famine/outbreak were (literally) not happening anywhere in the galaxy for a while.
OP, in short; Influence does nothing for you unless you wish to change some labels or colours on the galmap.
I'm sure the changes are much more complex than that but other than peering under the hood, playing the BGS really doesn't change more than that which is... Unfortunate for those who want to, especially considering how dedicated and clever they are about it. That said, I posit a related question; with the natural increment towards famine/outbreak, would the galaxy sans any player interaction at all eventually deteriorate into a humanity-wide famine/outbreak? Or are there natural corrective ticks within the system? It seems like a ticking time-bomb otherwise.