to Random and CSIfemale
Thank you for your explanations. But here's why I think you are wrong.
You have entirely underestimated the effect of social conditioning, which emphasises some characteristics in girls and other characteristics in boys. Although you have mentioned it, you have suggested it is only part of the effect. But it is so powerful, and so prevalent, and we are not conscious of it because we cannot extricate ourselves from it.
Arguments based on 'evolution' are baloney, because they start with a backward projection from present day society, then they use that to 'explain' present day society. EG; the argument starts with the notion that women are the stay-at-home nurturers; so it is suggested that primitive societies had a category of stay-at-home nurturers; and that is used to justify the starting point that women are stay-at-home nurturers.
If you think about it, it is inconceivable that primitive societies could have had a group of people who just stayed at home, sweeping the cave and tending the fire, because the process of finding food is so energy intensive. Most of those societies must have been on the verge of starvation for a lot of the time. They could not have supported members who were not actively engaged in providing for the whole group.
And as to 'leadership qualities' - remember that the early British tribes had female High Priestesses, female warrior leaders, etc - mostly eradicated when the Roman Church took hold in Britain.
CSIfemale you suggest that animals have different personalities according to whether they are male or female. Please point to any research that establishes such a thing. Certainly there are different behaviours - all around courting and reproducing - but across all the species and within species these behaviours are not consistently performed by males and females. EG penguin males looking after eggs and chicks, male seahorses raising offspring, etc.
Within the animals where it might be possible to observe something that's more like a personality - eg domestic dogs and cats - surely you aren't really suggesting there is a consistent tendency for male dogs vs female dogs to have different 'personalities' or to behave in particular ways? Each of them can be dominant or submissive, aggressive or docile, etc.
But the chief and central point is this. Biology is binary. Personalities are not. Even if one accepts your view that there are tendencies across large groups of people for certain characteristics to be more prevalent in men or in women, 'Gender' is not a helpful concept because it tries to force people into boxes or requires them to reject the boxes and declare themselves non-binary gender or gender fluid. Why have the boxes at all?