However, it doesn't explain how they (with no facilities in direct control) can suffer from an agricultural blight, while other system factions (including controlling faction) do not. What kind of blight is this?
I think there's an expectation here that the BGS is far less high-level and abstract than it actually is. What ownership of assets means, what influence is, what status effects mean, etc. has a lot of potential detail that isn't defined (and wouldn't necessarily be particularly relevant to flying a spaceship if it was)
It's obvious - especially in the high population systems - that there must be thousands of settlements and facilities not visible from our view (whether on non-landable planets or just too small to consider) simply to fit everyone on the population number in.
Similarly just because the economy is listed as Extraction doesn't mean that there is absolutely no local food production whatsoever - just that the system needs more food than it can purely produce locally. Likewise many non-Extraction systems you can see mining going on in their ring systems - just not at an intensity that means they don't have to import it.
(A lot of this abstraction is of course essential for gameplay purposes - in the real world, economic effects like booms and busts, conflicts like wars and elections take place over months or years rather than a week. Speeding it up to a pace that makes for any sort of gameplay means agreeing not to ask too many questions about certain "realism" aspects of it...)
Ian sums it up far more succinctly than i could.
There's a lot of "missing detail" in the BGS, and so it's literally a "fill the blanks" exercise.
Extraction economies take fruit and veg as am import. Presumably that gets sold to organisational entities which act as distributors, who generate that market demand, with the facility controller taking import tariffs etc. which is how that generates influence for the controller.
Those distributors comprise aspects of the factions. It's not unreasonable to think that a non controlling faction could take receipt of blighted fruit and veg, which affects other stocks within that distributor, while remaining sufficiently separated from other distributors, shielding them (And by extension, other factions) from similar blight problems.
Ian also hits a good point with food manufacturing. Grain is also an import... but I'll almost certain individuals on a station just buy grain and eat it... it would get used to product flour, and subsequently bread or other grain based products. If an import shipment of grain was blighted, and it spread to other organic products, it'd be disastrous.
The next question based on that would be "but why doesn't other factions suffer blight if i sell them blighted goods?"... well, because the blight is known, as well as the origin of the goods, and so they can quarantine those goods until they can deal with the blighted stock.
This is all just blank filling, and not codified I the game, just one interpretation.
