There's a false equivalency at play here between racism and sexism. All white people are racist because the power structure gives no advantages to black people whatsoever. There are, however, very specific spheres in which being a woman comes with certain benefits. These are few and far between, but they do exist. Courts favouring women over men when it comes to deciding who the children will live with. The rate of homelessness is proportionally higher for men. The rate of suicide is higher for men etc.
Because there are some advantages conferred to women the label of sexism is valid, though I'd be loathe to suggest using it outside of these specific instances. The great RBG and the Moritz case (1972) is a good example of sexism, and the more recent Rotondo vs JP Morgan (2019) also applies.
An attempt by both sexes to advocate change to lower the suffering of both is the goal of feminism - but that doesn't negate the fact misandry exists too. That might be justified based on the personal history of those pushing for it - but counter-intuitively damages the goals of feminists overall.
The difference between #MeToo and #BelieveAllVictims is an important example. The first was about educating males about the lived experience of women, the latter was about attributing guilt at the point of accusation and bypassing legal process. The first encouraged refletion and meaningful conversation, the latter caused defensiveness and a closing down of the conversation. That led directly to distrust between the sexes rather than empowerment of both.
What is required is that men and women work together for an equitable and safe society that benefits both. It's about empowering each other. That's not a quick fix, but this isn't a zero sum game. It's less likely to be achieved by each side preaching to the converted in an echo chamber. I'm not suggesting that you're doing that - because I think you're smarter and a much better writer than that, I'm saying this is where the discourse is headed and due diligence is required by all moderates and centre ground men, women and penguins.
