Back towards the main topic more . . . I think the OP does have a point, but we are missing one important thing: Context.
The statement given by the OP is relatively innocent and laudable in the context of, say, educational opportunity. The same statement takes on a more sinister tone when in the context of, say, reproductive health.
The article itself on Wikipedia seems to talk more about not forcing stereotypes onto people, which I find acceptable, than about trying to eliminate reproductive differences (BC and abortions). I know my DH would appreciate a “give up your seats for parents carrying young children” campaign on public transportation, as a man who rather frequently headed out on public transportation with two toddlers. While boys and girls do tend to play with different toys, exagerating the natural tendency by turning it into a false dichotomy (boys play with cars, girls play with dolls) isn’t appropriate - some children will naturally play more with the other toys, and that is fine. There is mention of lighting streets at night - a policy geared towards giving women equal opportunity to walk and exercise at night, since women have reason to be more concerned about night-time safety than men do.
Where I see the risk is when we start trying to eliminate the inconveniences of pregnancy and birth by making it easy for women to avoid them through BC and abortion at any cost. As a society, we seem to want to avoid the reality that women are inconvenienced by childbearing (pregnancy, birth, and nursing) for several years per child, and rather than adapt society to support that reality, we’re trying to force reality to fit a warped image of society where women don’t have those inconveniences.
The other place where gender mainstreaming is an issue is when negative “male” behaviors are falsely seen as a healthy norm and better than “female” behaviors. I’m thinking specifically of “promiscuity” and “chastity”. Society has long found male promiscuity more acceptable than female promiscuity, and female chastity more valuable than male. There is nothing wrong with destroying this unfair stereotype, but society took the wrong path; instead of normalizing the healthy “female” behavior (chastity) for both genders and valuing male chastity, we normalized the unhealthy “male” behavior (promiscuity) for both genders. I used quotations because those behaviors were never actually male or female - God made both sexes for chastity.
If we as a society want women to be more active in public life and the workplace, that’s fine. If we want fewer double-standards towads men and women, that is also fine. We just shouldn’t do it at the expense of uniquely feminine responsibilities, nor should we assume the stereotypical “male” behavior is always best.
Satan loves to attack good things, both by twisting something until people think black is white, and by trying to convince us to throw the baby out with the bathwater if we try to get rid of the twisted perceptions we now have. Equal opportunity and a stronger female public presence are good things, but trying to eliminate the periods of vulnerability that women experience due to childbirth by removing childbirth, or evening the sexual stereotypes by making women permiscuous, is a twisting of what is good.