The benevolent trait that does a one time flip does flip Minors, which otherwise (I think) would ordinarily not flip due to ordinary influence.
It also grabs starbases, which you normally can't grab.
else, the math for influence is also questionable (not just the AI). I don't know the current math, but it appears that Influence only grows over time and it's a contest to see who can grow it faster, with no downside to deleting influence buildings once you get 'enough'.
That in and of itself is a problem.
The game mechanic really should be similar to --
Inf range of planet (Ie, what's the maximum distance this planet can affect)
Inf strength of planet (how much Influence does it produce)
Each hex (starting from planet) gets a (planet strength / distance from planet) value out to planet range. (PS/1, PS/2, ..., PS/PR)
Where Planet range is mod'd to a 1-100% range. (1 hex from planet = 100%, max range from planet = 1%)
Sum all Inf from all sources.
Contest vs other influence producers.
Simple and straight forward. Plus, if you modify the range or strength of the planet (by adding or deleting improvements/upgrading them) it modifies it on the map right away.