Neanderthals had bigger craniums. There isn't much evidence to suggest that they had more cognitive abilities. And if you're going to discuss gender differences, reducing it to the two most typical sex genes is still unsound when there are several others that have and continue to exist especially when we as a society define it by body parts and few actually know for certain what sex chromosomes they have. If you want to bring evolution into it, you need to look at all sex gene types that can continue the species - which is far more than XX and XY - and somehow cut out our current social binary in how people beyond those two interact with the world.
All the evolutionary psyc talk reminds me of archaeological sciences about 50+ year ago (or any on TV now) where anything that couldn't be explain was explained as a 'ritual' because that's how they saw earlier people. As a concept it may have merit but in practice it -- and you - are taking fragments to fit a worldview which isn't really how good science is done. But then, science is only as good as the people doing it and interpreting it as anyone with any grasp of science history knows.
We have no idea why the male monkey babies preferred the mechanical toy over the plush. It could simply have been because it was shiny or harder feeling or anything. They could have been influenced by the workers - anyone who has done any research with people or other animals knows how easy we all are to being influenced. The very people wrote it as has been pointed out cautioned against using it as you and the media keep doing.
About American Indigenous nations, the over 600 nations we have records, most of which did not have wheels prior to European colonization is that the differences in tools and choices in technology created can suggest what is more effected by evolution, what is more effected by environment, and what is more effected by culture. Of the largest American Indigenous nations at the time of European colonization, the biggest technologies were mainly in foods (of which most popular parts of the Western diet now rely), landscape shaping, soil maintenance that we still cannot replicate, and many other non-mechanical technologies.
You began by stating that your sons were interested in wheels and not your daughter and that wheels, along with other technological advances, were created because of men's testosterone. Except several hundred nations for tens of thousands of years did not and most of their technological advances which many enjoy today aren't mechanically based as you described. They were not "primitive" [seriously?], at the time of European arrival they had some of the largest cities in the world and the Americas had large, long standing trade routes, there was just nothing that in the environment or culturally for such things -- and apparently nothing in evolution either. There is also no evidence that I know of that suggests that tool making anywhere, even in animals, is strictly a male thing and nothing to suggest that any of these early tool were originally invented by males. That is a leap you are choosing to make. We've already seen in the last decade or so many cases of well known early tombs or constructs attributed to men because of the weapons and armor and tools that turned out not to be so - the people then just assumed so because that is how they saw things and chose to make that leap. They were wrong.
Like archaeological sciences of old/TV, evolutionary psyc today suffers badly from echo chamber syndrome and some of them and especially the media have a lack of humility in admitting that they do not have a clue what something means or that they could be wrong. But then, some of the biggest research universities in this area are still funding research and doctorates into finding which race is "scientifically" smarter or more attractive so maybe it needs someone to unstick them from 50 years ago before it will make real progress. Until they can move beyond that, the evo-psyc that makes it to the general public is better viewed with a high amount of bollocks detection.
In my house, we have a box of nerf guns and crossbows - my partner and I collect them and keeps the ones we won't use or he wont paint in a box for the kids. It goes untouched except when another family who have four girls come over and make everyone play with them. None of my four have interest in wheeled toys except to decorate their block towns. All of my kids love dress up, and building, and electric experiments, and video games and swords. We have an extremely large plush toy collection. My 2 sons are less flexible about what they choose to watch than my 2 daughters - they have their favourites and nothing is going to change their mind without a sulk while the girls are more likely to experiment and try new shows but any of that could just be them rather than their assigned gender. Really, even with kids from 11 to 5, how many times I have to request they take their hands out of their pants is pretty much the same across gender. I notice people laugh and talk about it more if they catch me telling my sons compared to my daughters, but they all do it. Messing with genitals I can see the argument for evolution, but I find it harder with their TV watching habits or lack of interest in wheeled toys.