Lewontin's Fallacy on Gender

Lewontin's infamous argument stated in a not so strictly accurate analogy boils down to this:

Since the height differences amongst the groups men and women are much more than the difference between them(4-5 inches) the groups themselves don't exist.

Lewontin's conclusion stated for gender would be:

“Human sexual classification is of no social value and is positively destructive of social and human relations. Since such sexual classification is now seen to be of virtually no genetic or taxonomic significance either, no justification can be offered for its continuance.” 

The first sentence being implemented since the 60s culture revolution to an unsurprisingly devastating effect, the second however losing its meaning since genetic difference of sex chromosomes is not as easy to refute(and a big part of why this analogy is not 'strict').

Stated this way, one could easily see that the problem lies not with the grouping but the criteria that is being used and how it is being applied. A feminist rebuttal to the above can be that since it is gender that is socially constructed while sex is a biological fact, the fallacy should be written as :

Since the height differences amongst the groups human males and human females are  much more than the difference between them(4-5 inches) the groups themselves don't exist.

Gender classification still sounds wrong. For biological things like body-shape, hormone levels that are responsible for it, fat-content, muscle mass, etc., the term used should be sex and not gender.

So how does one show that genders don't exist? That gender is nothing but a social construct?

Ms. Hyde's monumental study of male-female differences in personality, with the Gender Similarities Hypothesis comes to the conclusion that:

males and females are similar on most, but not all, psychological variables

The work has been cited 924 times and has won Ms. Hyde not only media time but also awards in her field (her other paper regarding gender/sex differences in mathematics and science, or rather the lack thereof has been another feather in the bonnet). According to her award-winning, media-inflaming and androgynous gender warriors' go-to scientific literature when it comes to internet debates:

78% of gender differences are small or close to zero

and 22% aren't? mmkay!

Her hypothesis is recently argued against much in the same way the Lewontin's argument was criticized in the paper Lewontin's Fallacy by A. W. F. Edwards in 2003. That is, instead of taking single criteria(univariate modeling) on which there is considerable overlap between groups, they make use of multiple criterion to differentiate between two groups(multivariate modelling). From the 2012 paper 'The Distance Between Mars and Venus: Measuring Global Sex Differences in Personality', the authors come to the conclusion that:

....we believe we made it clear that the true extent of sex differences in human personality has been consistently underestimated.

It's funny to note that what Ms. Hyde said in her paper with the above proclamation in mind:


The Gender Differences Model, which argues that males and females are vastly different psychologically, dominates the popular media


Curiously she also makes the statement that:

"self-esteem is roughly as much of a problem for adolescent boys as it is for adolescent girls"

So apparently AAUW study on how schools shortchange girls was wrong or a statistical outlier(they accidentally the whole data!) or maybe it has brought equality to this sphere in the space of mere 20 years!
Since feminists are never wrong and since patriarchy has been known to keep women in low spirits, the only reasonable explanation is that equality has succeeded. Girls have overcome the trauma of becoming women!
Progress, comrade! Human nature has been modified, estrogen shock has been minimized and testosterone has been reined in, Vive la révolution!

Or boys are now as neurotic as teenage girls, still who cares! Equality! Equality!


PS-

According to the American research, there has been a 67 per cent increase in [Narcissism] over the past two decades, mainly among women.