Good example, although not neurological, is the military. Men are on average stronger than women, which is a desirable trait in the military.
Pain resistance and cooperation are also desirable traits in the military, often moreso than physical strength in a modern army, and women on average have the advantage there.
Possibly a more practical thing (not for the fleeting desires of society, but its longer term potential) is that if a male soldier dies, his contribution to posterity can be fulfilled equally well by
any surviving male soldier at all. And, at least until it becomes a ridiculous amount, can double, triple or even far more his contributions by filling in for as many dead colleagues as are unable to return (or return 'intact') to fulfill their peace-time duties for the stay-at-home women of the nation who would be necessary to ensure that there's a future generation of the nation. And can even do their bit of non-military 'service' on a weekend pass or equivalent home-rotation. (Or, frankly, via any small container, properly handled.)
Whereas, with an Amazon Army, each battlefield loss narrows the options for the then typically nine-month-and-more bottleneck on home-front replenishment[1] and that may not be a happy situation, before even considering the excess of men[2] with far less 'war-work' to do, away from the more direct slaughter. And so clearly a standing army is more practical (in the long run) to be all-male than all-female.
Which is not to say that a mixed-membership service is less optimal than your tried and tested female-barring versions, either, and (even prior to full Brave New World attainment) not all of those nine months need necessarily be fully excused from at least support roles. But it
might suggest that a battalion or three of post-menopausal recruits could swing the advantage of a war, if it gets that desperate, with few disadvantages and possibly a number of additional advantages. (Assuming you can tactically coordinate the perhaps useful mood-swings, etc[4]...)
[1] I know there's more time needed than that, but
anybody could technically take over from that point. With 'uncles' and 'aunts' being logistically just as interchangable with whatever parent(s) you divert back to the front. If not fully amortisable into one-to-many creche-'commanders' who then instill an age-hierarchy over their young charges to streamline and regiment the process of reaching battle- and/or birthing-age themself which it sounds like this hypothetical perpetually-at-conflict society is going to do
anyway, right?
[2] Assuming somehow men aren't otherwise infrequently available, and valued highly for every bit of contributive effort they can be pursuaded to give back at the 'farm', as such a precious commodity[3] to the female part of the population that it is now they who have to go into the martial sphere rather than have the option of choosing a marital one.
[3] I say... is this because of something back in the mists of time, much much earlier on in this multi-generational war, that was unleashed upon the region (nay.. the world!), somehow targetting the Y-chromasomes in a devastating (and possibly uncontrollable) strike against the roughly-parity but disproportionately armed populations of the instigator's enemies? I really think we need to know! And, naturally, it managed to cross back to the very place that developed it, too. Because of course it would.
[4] j/k! ...in case that needs to be said about much of this post. Once into the speculative hyperbolae, certainly.
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