I have not read the book being reviewed, but I don't see any radical or ideological claims in the review. The main claims of the review appear to be:

1. Articles have been published which claim to identify male-female brain differences.

2. Citing a few specific examples and claiming that the book delves into more, these articles are typically wrong due to innumeracy, bad designs, lack of power, etc. I assume good faith that the criticisms she identifies of the studies in particular are reasonable, mainly because the kind of innumeracy and power issues she identifies plagues all the research I read in other fields and I have no reason to believe quant training or publishing incentives differ materially in neuroscience.

3. Without saying the articles are sexist, the conclusions drawn from the articles have demonstrably been sexist in the past (i.e. moving from claimed differences in grey matter to generalizations about women's aptitude in mathematics)

4. Because we're fairly good at no longer saying women are worse than men biologically, we now say women are different than men biologically. The studies that report this have, according to the reviewer's summary of the book, similar problems.

5. If we assume that the research on biology is incorrect, how do we square that with apparent differences between men and women? Cultural arguments, for which studies are cited.

Again, it is quite possible I am missing the anti-science backdrop of this debate that you seem to be familiar with, but the article you linked does not exhibit those kinds of qualities. It is of typical quality for a review article. The claims made that are heterodox to "mainstream" thinking appear to be grounded in criticisms of science from within science. When I say "much of social psychology is wrong", it's not because I'm an anti-science Freudian or whatever, it's because I'm a scientist and I'm numerate and I know much of social psychology is wrong due to analysis errors, researcher degrees of freedom, fraud, file drawer problems, publication bias, etc. I mean, probably the most direct claim examined in the review article is the reviewer claiming a relevant study found one thing and a subsequent, better-powered meta-analysis found the opposite thing. This is not an anti-science howl, it's literally what we want scientists to do.

Do you think you could do a better job of highlighting the ways in which you feel the article reflects a "flat earth movement"? Again, I'm at a disadvantage here because I haven't read the book being reviewed, but since you have, evidence from the book would be useful too.

I mention this because the Medium article we're all replying to seems to provide... uh... exactly zero references to studies, numeracy, good scientific practice, etc. It seems mostly founded on incredulity that anyone would make the claim you pull-quoted -- actually, even worse, it seems mostly incredulous about the TITLE of the Nature piece. It talks about Gender Studies, and culture war; it repeatedly mentions Quilette but I don't see the mention of any scientific journal after the Nature review piece. He does cite his own experience from all the studying he did.

One of the claims about Gender Studies is that methodologically it, like many cultural studies programs, elevates lived experiences over empiricism. If you're an empiricism, this is scary. But the Medium article has the professor claiming that he knows differences are obvious because he's seen them himself, and so the article which actually discusses empirical results must be wrong...