Secondly, the author doesn't make a convincing argument that the fact that the reason why men get all the risk and all the reward is because of something innate, instead of a self-perpetuating social system that actively encourages one gender to risk it all and reap the rewards, while holding back the other gender to mediocrity and risk-free existences.

The author may not have gone into it, but the argument exists, based on what we know about genetics and reproduction.

Female reproduction is inherently limited, first of all by the time it takes to have a single child, second by the probability of death due to childbirth (relatively small, but not insignificant for most of human history), and third by the reduced fertility that comes with age.

Male reproduction is essentially unlimited; the potential maximum "genetic fitness" (measured by simple count of the branches you spawn on the tree of life) of becoming a king and impregnating an entire harem of women over your life is easily 10x the maximum "fitness" for a woman (though the probability is very low for such extreme situations), so it's to be expected that any preferentially male-expressed genes that would increase the probability of ending up in that situation would be more prevalent in the gene pool than the corresponding traits for women (which would be mostly neutral evolutionarily, since a woman with a male harem is not going to spread, on average, more copies of her genes than she would in normal life).

We don't need to assume that the dominant factor is the king + harem situation, either; it's enough merely that men, on average, see wider variability in reproduction than women do (cheating, cuckolding, etc. tend to make that happen).

The "leaps of faith" required to let this explain increased male risk-seeking are:

a) That the traits we associate with risk seeking correlate well with the traits that cause men to be highly genetically successful

b) That the way things actually transpired in history, the variance in male genetic success was, in fact, significantly higher than the variance in female genetic success. Note that we are not talking about averages (they're equal, quite trivially), but variances

If you accept a) (which is not much of a stretch - sleeping with other men's wives is definitely a risk-seeking behavior that increases genetic success for men), then it's absolutely 100% certain that the amount of genetically linked risk-seeking will be higher (or equal) in men than in women. It's very easy to make a similar argument that implies that any genetically linked risk-avoidance genes that are preferentially expressed in females will be more common than those that are expressed in males, at least to the extent that they would reduce male reproductive variance without a corresponding survival benefit.

Bear in mind that your comment about a "self-perpetuating social system" and the things that it encourages may not be entirely off-base, but that doesn't change the genetic imperatives: given what we've seen above, such a social system would align very well with the genetic best interests of its constituents, so it's hardly a stretch to imagine that the two factors have coexisted and reinforced each other quite strongly.