Maybe it's just cultural. I don't see why risk taking favors men millions of years ago.
But in terms of genetics the female upside is limited by the very high costs of child bearing & the limited number of children they can physically have. The male genetic upside to risk is limited by the number of women they can have sex with.
1 / 200 men are apparently descended from Genghis Khan! This feat would simply be impossible for any woman of that era.
Having many low investments into offspring can be effective, but it is also risky. The more champion based a race is (one male per many females), the more energy is going to be spent on competition. The offspring will also have to depend solely on the mother, since each child is just one of many from a single male and has thus less individual investment value.
The alternative strategy is to focus on a smaller number of children, making sure that that the reproductive investment has a higher chance to be successful. Children gets the value of having two parents that are invested in the offspring, but there are issues of "cheating" (an observed behavior in many animals).
Of course, I am only describing male strategy here and naturally it can not exist in a vacuum. Female strategy often has higher initial cost, but it does not dictate the outcome. In the case of humans also, the observed strategy in both sexes is normally said to be a mix strategy, so it is far more complex than just saying that the male genetic upside to risk is limited by the number of women they can have sex with.
Non-humans do not I think enter into the discussion, and a strategy doesn't have to be universal to have a big enough effect on our psychology to be worth worrying about at the level of education policy.
Risk behavior is complex concept, goes well beyond reproductive strategies. Behavior is significant more effected by culture than genes, and culture is effected by gender identification rather than sex.
Would you be surprised if transgenders has a identical risk taking behavior in relation to the gender which they identify themselves as? Would you be surprised if sexual presence has no impact at all? In both cases it would strongly imply that reproductive strategy has no impact on risk taking behavior.
Some explanation for the claimed propensity of men taking higher risk would be that being successful is more rewarded socially for men then women, while averageness and failure is generally socially more for men than women. Finding root causes for that social pressure is harder, but my guess is that it has both biological reasons and historical ones.
Even if so, I'm not sure if he would actually be an example of the low-investment male mating strategy. He was the ruler of a large empire, so he would be able to invest in his children. Even those who weren't recognized would have advantages from being rumored to be his offspring.
Groups which allow the low-investment strategy are outcompeted because they don't incentivize any males to invest in the group, while the Mongol empire had plenty of conquered people available to the Mongol men who were excluded by elites' polygamy.
Civilization is very recent in human history, and without civilization you can't have Genghis Kahn so I don't see how he is relevant.
I feel like people used to say God made people a certain way. Now that I basically only interact with nerds, they just replace God with evolution. I'd prefer some citations, since all I hear are the same assumptions.