The wage gap exists because the average woman has major career distractions: children. Average men have an edge because they are able to invest more time in improving their value to employers. The video's speaker seems to have a problem with this, and proposes that men should take on more household duties to balance things out so women can put more into their careers.
But why should "closing the gender gap" be a concern at all? I see no reason why the statistics must be balanced. Indeed, it makes more sense to be unbalanced... it's called specialization- which increases our prosperity. Because women have the children and must be closely involved with them when they are first born, it's natural the women will specialize in running the home while men will specialize in bringing in the dough.
None of this diminishes the value of women (or men) one iota. We are not valued by our income. Needless to say, women are and should be equally free (legally) to pursue careers as they wish, and misogyny should be eradicated (it is for the most part), but beyond that, this is a non-problem- it's often a good thing. Many families find it works best for them for the man to focus on supporting the household while the woman focuses on running the household.
Let's spend our energy focused on real problems. To bad "Learn Liberty" is getting distracted.
Related Post: The Fake Equal Pay Crisis