Well, seeing as their report is supported by 400 scientific references, they have a legitimate right to exert some influence based on its findings imo. That influence is not likely to be decisive given the overall conservative nature of US politics.
Getting back to the Swedish example, which is more interesting to me because it shows what actually happens when theory is put into practice as opposed to what opponents of all things liberal fear will happen, there has been a study (unfortunately not free to access) concerning the results of such policies. Here's a summary of the conclusion:
"While the conclusion suggests we’re genetically prone to immediately ascribe gender categories to others, it also suggests socialized differences can be mitigated. Compared to children from traditional preschools, children from gender-neutral schools were more likely to play with unknown children of another gender — an important finding since, “young children who favor same-gender playmates develop more extreme gender-typed interests and behaviors over time,” the study authors write, citing previous research. The children from gender-neutral schools also held fewer gender stereotypes (e.g., dolls are for girls)."
https://theswaddle.com/the-results-of-extreme-gender-neutral-education/
So problematizing gender categories and acting on that through education policy results, it seems, not in boys and girls not recognizing each other as different, but in:
1) More openness to other genders
2) Fewer gender stereotypes
So, getting back to the OP's hypotheses:
1)
To the extent that gender differences are biologically rather than culturally conditioned, gender neutrality in education and wider society will have no effect on personality or identity.
Seems to be borne out by this study re identity at least.
2)
To the extent that such differences are culturally conditioned, they are distorting constraints on human freedom, barriers to equality, and potential causes of psychological conflict and trauma.
More controversial. But if one accepts relatively less openness to other genders and more gender stereotypes to be distorting constraints on freedom, to be anti-egalitarian and undesirable psychologically, which seems a not immoderate interpretation, then also borne out.
So, if the fears that gender-neutral schools are a damagingly disruptive form of socialization that perverts our children's genetically programmed understanding of sex differences are wrong, and this form of education merely serves to undermine socialized stereotypes that are a hangover from a less enlightened past, should we not all get on board?