Thats an interesting take and not one I've heard before. I don't completely agree with it. Of course I'm also not a researcher.
It seems weird that limited food resource peoples would be able to devote both time and calories to other none food gathering tasks. Namely the women who are already having to travel farther for less food. Yet aren't dealing with the complexity of hunting game. Namely large enough game for the group. Not that it isn't possible. Also won't the production of pottery require kilns to harden the mud? Of course, assuming northern Europe, they would already be pretty good at building and sustaining large, hot fires. So that may not be that big of an issue.
Also why would male brain size increase more then female brain size, despite the women having a large percentage to reclaim or possibility thereof. Women's education over the past 200 years is been as good as men's education opportunities, but still far better then post-ice age tribes opportunities.
Typing quickly to save time as this is a lengthy post -- please forgive errors.
First -- did you read me correctly? I said that after the ice age women's brains reportedly underwent a much larger decrease than men's brains. Since we're more or less cognitively equal now, I speculated that it's reasonable to assume that women of the time probably had the advantage over men.
I'm not sure what to make of the education comment or what you meant that has to do with anything. Intelligence leads to people wanting education though; not education causing intelligence; there's a lot of literature on this but the theme is basically that there are no interventions that will increase IQ as measured in adulthood. Barring gross neglect, your IQ on a culture-fair IQ test should be the same regardless of whether or not you have gone to school, learned to read, etc. and stay more or less stable for life. In studies of adopted twins, they have very similar IQs -- much more so than between each and their adoptive siblings. Which is exactly what you'd expect of something that's almost entirely hereditary (within the fairly gentle conditions prevalent in our society -- no doubt it'd look less "hereditary" in a society where a large percentage have brain damage from neglect or toxins or whatever).
How familiar are you with modern hunter-gatherers? Throughout Africa, the men laze around when not hunting/fighting (cattle-owning tribes typically regard their care as the province of men but here even boys do most of the work). Women basically do everything else, e.g. gathering edibles, tilling the soil with a stick for the the very primitive type of farming they engage in, collecting water, raising and educating children, etc. The first time I saw it, the idea immediately occurred to me, and only half-facetiously, that these men really are the forefathers of the modern black pimp.
If this is typical of our mesolithic ancestors, it is not surprising that women would need to be cleverer than men. They have so much more to learn than men to be competent -- especially to succeed in miniature arms' races against neighbours. This is especially true in Europe. Unlike Africa, where death can arrive suddenly, randomly, and in a manner difficult to take precautions against, Europe is more unforgiving in its climate and conditions but the cyclical change of seasons, animal migrations, etc. allow more scope for an intelligent, prepared tribe that can delay its gratification and put stock away for the winter months to thrive. This is a strong evolutionary pressure for intelligence to increase. Dying of random diseases in a temperate climate where food is always more or less available (unless it's not) -- not nearly as much.
The brain uses 20% of our energy expenditure today. A very high amount -- maybe it was higher in the period discussed when our brains were larger still. Remember that any allele that leads to even slightly less success ("fitness") than another will become decreasingly common with the passing of generations until it is is eventually eliminated and the other allele is represented in 100% of the population ("fixation" is achieved): how quickly this occurs depends on the fitness differential between the two in some pretty simple algebra. Where populations have a small number of ostensibly fitness-reducing alleles -- e.g. the ones that cause sickle-cell anemia -- you need to either question whether they are truly fitness-reducing at all or serve some other purpose that makes their persistence advantageous (as in this case), or that they must be too recent in origin to have yet been eliminated by selective pressure.
So from the above -- I think it's pretty clear that evolution is not going to keep any more brain matter around than it strictly needs to. Intelligence is not an unalloyed good -- rats and mice for instance are two animals that evolve quickly enough to observe but we don't see them getting smarter and there's no reason to think they'd do better in their niche if they did, or at any rate, not enough so to justify the metabolic costs (and likely the increased maturation time). It's pretty believable to me that in the agricultural period after the hunter-gatherers but before the rise of large communities made specialization in trades a possibility that there was a long period where we lived comparatively very cognitively undemanding lives compared to the past. Unless our social lives or something increased in complexity to justify the need to keep such an expensive organ around, it's reasonable to think it would've continued getting smaller until doing so gave more disadvantages from decreased intelligence than advantages in terms of energy savings.
Evolution can also put different selective pressures on men and women -- we could have wound up way more different than we have. It could have put selective pressure on women to become bigger and smarter and men to become smaller and dumber; it didn't, but it could have. Human males and females are pretty close to the same size -- probably a sign that unlike gorillas with their much greater sexual dimorphism, men aren't in a huge degree of competition with one another to get huge and beat one another up and steal each other's women. We're not markedly different in intellect either -- maybe because there was an advantage in men and women being able to learn each other's work when required but more likely so that neither sex would fall to a disadvantage socially. Yet there are areas (spatial intelligence, an asset in hunting and navigation) where very few women can keep up with men; and women have a small but distinct advantage in verbal intelligence.
I mentioned the evolutionary pressure against an excessively large brain for reasons of energy consumption. Two other things come to mind: first, a baby's head obviously needs to fit through its mother's pelvis so any advantage in increased skull size is going to come at the detriment of women's ability to walk quickly and efficiently. There have already been a lot of adaptations to allow us to get so big -- skull sutures allowing it to get squished on the way out, huge fontanelles that don't close for years, survival of extremely premature birth (look at how helpless a newborn human is compared to a chimp!), Africans are even born a week earlier so they can emerge from their mothers' slightly narrower pelvises (big butts!). Second, the sheer heat generated by the highly-metabolic brain is going to impose limits on its size. Thermoregulation was no doubt easier in ice age Europe than in Africa causing different trade-offs to be made in the respective regions between size and intelligence. It's not too surprising that this decrease in brain size began as things started becoming warmer.
Brain size only has a 0.4 correlation or so with intelligence, i.e. it explains a lot but most of the differences in intelligence are probably not due to gross structural factors. The evolutionary pressure towards increased intelligence may not have justified the calorically expensive and rather brute-force means of just increasing its mass but there are more efficient or metabolically "cheaper" ways of doing this -- e.g. Ashkenazi Jews' increased intelligence probably depends on novel alleles for lipoproteins in the myelin to speed up signal conduction, for instance, and explains their susceptibility to lipid storage diseases. There's no doubt our brains are more efficient than they used to be but obviously either the selection pressure for intelligence was not high enough to also keep the big expensive brains, or doing so was disadvantageous for other reasons.
Something like half of all genes are expressed in the brain and will cause some small cumulative linear effect (positive or negative) in terms of IQ. Only a few hundred are known so far but enough so that it's possible to work out someone's genotypic IQ that more or less aligns with IQ as determined by psychological assessment (by the same principle that you don't need to count all the jellybeans in a jar to estimate how many red ones there are if you count from a handful or so). The first country to take advantage of this for pre-implantation embryo selection for IVF, let alone gene editing, will end up ruling the world.