I don’t see <<claims about the “average” woman>> in Damore’s text. Please point them out. I can only find references to the traits of men and women “on average” and the “distribution of preferences and abilities of men and women” (p.3). Note the difference between for example “women on avaerage” and “the average woman,” and between the notiona of “distribution” and “individual”. In fact, at the foot of page 3, Damore unambiguoulsy preemts any idea or accusation of making claims about individuals by specifically stating “You can’t say anything about an individual given these population level distributions.” This is followed by a diagram on page 4 to which the author notes “Populations have significant overlap. Reducing people to their group identity and assuming the average is representative ignores this overlap (this is bad and I don’t endorse that).”
So not only is Damore not doing what you describe, but he also saw that ball coming and already hit it out of the ball park. As for rooting logic in reason (whatever that means) or indeed, rooting reasoning in logic (which seems more familiar), I see two logical fallacies at work in your reasoning here.
1. Falsely characterising the other’s argument in order to attack it is a classic “strawman.” He even preempted the false characterisation in his true argument, which suggests cognitive dissanance on your part if you even read his text at all.
2. Assuming that the other does what you do, reasons from the same premises as you do, is a less known (but very common) logical fallacy that is probably best characterised as “the thief thinks everyone steals.” In this case, reducing people to their group identity and assuming the average is representative is a premise of identity politics (when it suits a particular argument). Assuming that the other is doing the same thing (just with a content you don’t like) is an elementary mistake, but the fact that the other actively denounces this premise should be an indicator that he’s not basing his reasoning on it.
Interestingly, nr. 2 crops up all over, for example when Christians assume that atheists are “anti-Christians” or when a president claims that the opposite of “with us” is “with the terrorists” (i.e. “against us”) or when criticism of a so-called “diversity policy” is construed as “anti diversity.”