I'm not sure anyone's quite claiming that.
Just that if you asked, ahead of time, a question like "So, what would it take to convince you beyond reasonable doubt that there's bias favouring men over women in the academic employment market?", the answer you'd get would likely be pretty much exactly what this study found.
Of course, there are always loopholes, just like a sufficiently ingenious creationist can always find contrived explanations for why some scientific finding is compatible with creationism. The speed of light is changing! The aftermath of Noah's flood just happened to deposit corpses in the layers we find in the fossil record! The world was created with the appearance of great age! Similarly, we can find contrived explanations for why an identical-looking application gets such different assessments depending on whether it's thought to come from a man or a woman. They might be really worried about maternity leave, and choose to define taking maternity leave as a variety of incompetence! There might be differences in competence between men and women that make a big difference to scientific productivity but are completely undetectable by academic testing and unmentionable by faculty! There might be really big differences that everyone conspires not to admit to the existence of! Sure, there might. And the earth might be 6000 years old.
What it would take to show that there is bias favoring men over women would involve showing that men are more likely to be hred than women and that this imbalance in hiring rate is not justified.
I remember seeing a talk of the concept of privilege show up in the discussion thread on contrarian views.
Some discussion got started from "Feminism is a good thing. Privilege is real."
This is an article that presents some of those ideas in a way that might be approachable for LW.
http://curt-rice.com/quotas-microaggression-and-meritocracy/
One of the ideas I take out of this is that these issues can be examined as the result of unconscious cognitive bias. IE sexism isn't the result of any conscious thought, but can be the result as a failure mode where we don't rationality correctly in these social situations.
Of course a broad view of these issues exist, and many people have different ways of looking at these issues, but I think it would be good to focus on the case presented in this article rather than your other associations.