With Equality through merit increases a person and conversely a genders power and privilege. Given not everyone is altrustic, it means there will be a proportion of the gender that will act toxically with the same awful end point.
One book I haven't gotten to reading is "The Power" which covers a view into this if roles were reversed.
But in the here and now, your adage you found fits. What most folks don't get (or maybe do and remained muted) is that ways a woman dominates a situation is vastly different then how a man dominates it. So overlaying one's gender viewpoint will be like a broken clock, right maybe twice a day.
Life is hard and it sucks, but the bigger question is how we can make 1+1=3 in real life by working together, versus the current of 1-1=-1 ....