I'd be curious if it's merely an expression or a true thing about fish.
In reality, it is the guts of fish that rot and stink before the head.
Source: https://www.phrases.org.uk/meanings/fish-rot-from-the-head-d... The guts of fish will start to rot quickly and spread terrible bacteria and disease throughout the entire fish.
Source: https://www.reddit.com/r/Survival/comments/mv04em/is_gutting...Hope this helps. I'm not sure why you were downvoted for expressing curiosity about fish biology.
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.