"However, in hunter-gatherer societies most hunting (and nearly all warfare/fighting) is conducted by men, which is likely to be connected with the fact that men and boys universally are more likely to display an interest in tools/weapons, in things that look like tools and weapons, in inanimate objects (as opposed to people-type things like dolls) and in objects which are about physically manipulating and moving objects."
a couple of issues with this:
In hunter-gatherer societies, hunting is often a very small part of life, done more for ritual purposes than to actually sustain life, so it's difficult to see how evolution could have played a major part here. (remember, the hinge of evolution is that not having X makes you unable to reproduce)
Lots of gathering activities (some of them performed mainly by females) also require tools: for instance, termite-gathering, fishing, clothes production.
To actual hunter-gatherers, the whole idea that hunting is focused on inanimate objects would probably be incomprehensible anyway: everything we know about anthropology suggests that it is very much seen as being about a relationship with a living creature.
In many traditional early agricultural communities (e.g. in West Africa), the heavy work of farming, which requires tools, has mainly been performed by women, while politics and history (which are people-focused are the domain of men.