The tl;dr is that he takes a bunch of stereotypes about men and women and asserts they're biological/intrinsic, and spends most of the document using pseudoscience and weasley language in a preemptive attempt to couch any blowback. (Perhaps undermining the idea that this was a learning opportunity & that the author did not anticipate a strong response. He explicitly mentions in the beginning that he anticipated a strong response in public and support in private, go figure.)
If you're looking for a single "gotcha" quote you don't really find it because the document is a fortress of wordy CYA. I think the only thing to say is normative statements like "women like people and men like things because women are wired to raise children" is post-hoc justification of the roles society prescribe for men and women, not some kind of truth bomb that's too hot to handle. People use similar arguments to the ones in this document to justify differences in pay and representation between races as well. It's a well worn path in sophistry to justify the status quo.
In the interests of fairness, I'll highlight a part of the document I thought was fairly insightful.
> Feminism has made great progress in freeing women from the female gender role, but men are still very much tied to the male gender role. If we, as a society, allow men to be more “feminine,” then the gender gap will shrink[.]
I think framing this as "men becoming feminine" is patronizing and prefer instead "allowing people a greater range of self expression, with men and women able to take on roles and personality traits traditionally reserved for another gender if they choose to (eg, there's no reason to highlight men becoming more feminine and not women becoming more masculine), as well as being afforded the opportunity to adopt non-binary genders previously considered to be invalid." I think it's underappreciated how men's gender role can be stifling and how men and women should both support the cause of feminism to achieve better outcomes for everyone.