I do believe that differences between groups of people and genders exist. The science is quite strong, though obviously there is quite a lot of debate possible around just how strong these effects are and how malleable these characteristics can be.
Unfortunately I have also come to believe (based on observing trends like those I listed above) that our society is absolutely not ready to discuss this openly. Not even close. Research in this area should be borderline encrypted in dense absolutely clinical and neutral rhetoric, should be discussed in private scientist-only forums, and should never be given a "pop science" treatment. (Most pop science gets everything wrong anyway regardless of the subject.) Any repeatable study reliably showing evidence for some systematic group difference among humans that has any strong social, political, or economic implication should be treated the way we treat exact buildable plans for a hydrogen bomb. Such things are classified for a reason. The destructive yield could be similar.
As I said I was once more optimistic. Recent trends (and not just in the United States) have changed my mind. Some individual people can face these things and discuss them rationally, but society as a whole is not ready and exposing everyone to this discussion is tantamount to posting a simple recipe for a biological WMD to Facebook. That too would be just fine if we as a human race were much saner, wiser, and more mature.
I think the ultimate reason this issue is so hard is that inequality is in my opinion the darkest aspect of the human condition. Inequality is worse than death. Death is something we all face and it only happens once, but the inequality is by definition unequal and those cursed by a bad hand from the genetic lottery die every day. The placement of sentient consciousness by nature into unequal vessels is such a horror that it triggers us all and inspires every kind of defensive emotional reaction, including of course a desire to hate and discriminate against those that might seem less fit than ourselves to hide from ourselves knowledge of the bad cards in our own genetic poker hand. The majority of the human race can face death, but it seems that the majority of the human race is not emotionally ready to face the greater horror of inequality.
Judging a community based on perceived motives rather than words and action is prime ground for problem. For example the Feminism movement is often being accused of misandry, while men's right movement is accused of misogyny. This make it is impossible for those two groups to have a conversation about eliminating stereotypes or that people should be treated as if all was equal, even if both groups share the exact same goal and only differ on how to reach it.
Going back to this study however, from a practical perspective all that a bad p value means is that there is a need to replicate the study if we still want to reference it. That is what the null hypothesis means. The data is indistinguishable from a random processes. The theory that blind auditioning is beneficial to women could still be valid, but this study doesn't really say anything to make it true or false.
If its not clear, I do very much believe that humans are all equal. Even from a gene perspective the difference are tiny and influence only a few minor traits. From listening to professors in neuro science and primatologists, it is a repeating pattern that where we thought there were biological difference between men and women that caused specific behavior, when tested it usually show that it almost always is caused by cultural factors. For example, men are not more prone to violence then women. Their higher physical strength and cultural preference to using their hands results in statistically higher damage, but the incident rates when studies tend to show identical numbers for men and women. While I have not seen a study to show it, I would expect to see the same result in regard to nursing and parenting. Breastfeeding causes a initial bias where women will statistically spend more time with the child, but beyond that it is all cultural.