I disagree that the idea you described is misogynistic because "women are better child rearers" is a positive attribute; men are painted negatively in this aspect. The result in this instance negatively affects men while benefiting women. That's misandry. Yes, it stems from the patriarchy. That doesn't change anything I've said. And you've failed to address how misandry is "inconsequential". Just because it's a smaller problem doesn't make its discussion worthy of ridicule or dismissal.
You cannot put a group on a pedestal without negative consequences. It may be “positive” but that doesn’t mean you should assume that. Misandry is inconsequential because misandry is primarily caused by women whereas misogyny is primarily caused by men. These things happen because of systemic issues, and the majority of people who run the system are cis-het white men. They may be hurting themselves, but they do it because of traditional gender roles. Misandry isn’t a systemic issue, misogyny is.