I'm very new to the game, but I believe the tech you're looking for is Research Focus (and later, Enhanced Focus), which gives your Trade Posts the selectable ability to aid Resource gathering.

Advent trade posts can (when the appropriate tech has been researched) switch to a 'resource boost' mode. When this mode is activated (click on the trade post, then on the ability), the trade post ceases to function as a trade post and instead boosts the rate at which extractors in the same system gather resources.

Note that this effect doesn't use refinery ships, and only affects extractors in the same system. It's also worth bearing in mind that you can have trade and resource boosting at the same planet by building more than one trade post, and that you can create groups of trade posts, assign them to a number key, and be able to switch between trade and resource boosting according to your current situation and strategy.

As to why Advent extractors have a 'refineries served' variable, can allies send refinery ships to each other's extractors? If not, I'd assume it's because it was simpler to use the same extractor code for all factions than create different versions just to remove a stat that the Advent doesn't use.