I don't think schools can do much here. MRA ideology is all over YouTube, which is the world's top streaming platform. Nothing that teachers say is going to counteract this.
I think a lot of people have spent years saying that children don't need fathers. People say this to let absent fathers off the hook, and to reassure mothers who feel guilt that their children are growing up without a father. But it isn't true. Reddit is full of men who are brimming with resentment at having grown up without a father, and felt stifled with a single mother.
The problem is that whenever anyone tries to discuss this, people go on the defensive. They immediately jump to strawman arguments such as "Are you saying it's better for children to grow up in a household where their father treats their mother like dirt?" But I don't think anyone, except for religious extremists, actually believes that.
There is no more effective positive male role model than a good father. Imagine if a gay male couple had a baby by a surrogate mother, and dismissed concerns about how she'll be affected by the lack of a mother with "It's okay, she has a female ballet teacher/netball coach." Most people would find the idea that could substitute for an absent mother laughable.
I think ultimately there needs to be much, much harsher social condemnation of fathers who walk away from their children. It should not be seen as okay in any circumstances. The step-parenting board is jaw-dropping: a lot of people seem to think fathers essentially become "uncles" when they split up with the child's mother, and are "helping" their ex by taking care of their own kids. Even if a woman lies about being on contraception (which is obviously an appalling thing to do) men should accept that every time they have sex with a woman, they're risking becoming a father.
Having more fathers stick around isn't a silver bullet, it won't end misogyny - but it would be much more effective than anything schools can provide.