I do agree, in German we have an idiom "Der Fisch stinkt vom Kopf her" ("a fish rots from the head down") to describe precisely how bad management cause ripples all the way down. That being said, CEOs are still prime candidates for automatization. I mean, surely nobody wants to replace good CEOs with bad software but rather with equally or better performing software equivalents. Additionally, one day it might be easier to ensure an AI CEO makes better moral/environmental decisions than his human counterpart.
At a guess (a wild-ass one, I admit): Some of the softest, most porous tissue of a fish, with the highest surface-area-to-volume ratio, and most exposed to the environment with all its microbes -- as opposed to most of their flesh, that is contained within skin and scales -- are the gills. So it feels very plausible that the gills are the first bit of a dead fish that would start to rot.
Well, those or the guts, of course: Assuming fish have an intestinal microflora like land animals, their guts would have a head start on rotting (Heh!) over even the gills. But I guess that's where skin-and-scales-as-container comes in again, only the other way around: the possibly rotting guts are hidden away inside the critter, so not as immediately noticeable as any rot starting from the gills up at the head.
This is all my private speculation, though, and most probably influenced by (i.e, my self-rationalisation of) precisely the saying you're enquiring about.
