It sounds like the author may have hit on something without really even fully noticing it. I'm sure it's at least something of an exaggeration, but she said she wanted to shove a bishop up this kid's nose because he was being a bit boisterous about taking pieces when playing against her daughter. She loves chess, but aggressive competition is bothersome to her. As she says, this probably has something to do with a lot of women not sticking around.
I'd imagine she probably sees chess in a pretty different way from that kid who was playing against her daughter. She probably doesn't loudly clink her pieces together or thwack them down onto the board when she's excited about taking a piece. She probably doesn't typically think of chess as violent. That's a perfectly valid way to enjoy chess, think about it, and play it. It's also perfectly legitimate to think about it as a strategic war game and to get a bit excited. It might be a bit uncouth for an adult to make a bunch of noise slamming things around, but for a kid it's not like weirdly aggressive behavior.
It seems to me that the conflict here is that while all sorts of ways of thinking about chess are legitimate, the predominant form of youthful masculine competition involves a sort of playful aggression that girls don't seem to be nearly as fond of as boys. Given that the majority of chess players are male it's difficult to avoid this and still play chess in large groups, so girls (and thus eventually women) continue to be underrepresented in chess.
It seems to me that the solution to this would be more women coaching chess and chess clubs directed at girls. Maybe if a girl's chess clubs were common there'd be less of a need for women's chess tournaments.