TLDR: "Power" does not equal "control". It equals... something more subtle that isn't quite control, and its directly liked to the person in question, rather than the superpower which that person might be aligned to. And yes, the Powerplay system uses the word "control"... but it doesn't literally mean that.
To give an example using your star system: suppose you fly into the main starport in this system. The starport is Independent-aligned. But it's Aisling's portrait flying from the holo-flags. She's their hero, even if she's not their direct ruler. And both the local authorities and the Aisling gang seem to be content with this arrangement.
Why is it like this?
Well, back when the game was launched, Powerplay wasn't a thing. The characters currently in Powerplay might have been mentioned offhandedly in news articles but would have zero impact on gameplay.
Players can freely join and depart from supporting a Superpower, virtually consequence-free, though their affiliated minor factions, but didn't have any way of directly "pledging" allegiance to a superpower. Players thought that "pledging" ought to carry more consequences: rewards for pledging, and punishments for betraying the team.
Frontier didn't want to completely scrap and replace the faction-superpower system, so instead they added a third level in between the local factions and superpowers: thus the Powerplay Powers were born.