First, I sharply reject the fallacy that racism and/or sexism is (supposedly) based EXCLUSIVELY on systemic/institutional factors. So do millions of other people. Otherwise, we wouldn't specifically refer to racist systems that oppress BIPOC as "systemic racism" so pointedly.

So, if you're using that as your operating premise...you're simply never going to persuade the masses. "Prejudice" implies a passive belief or attitude. Once that belief system becomes actionable or behavioral...that's where bigotry and discrimination rear their ugly heads to become a full-fledged "-ism" (I apply this premise to all categories of identity, not just race and biological sex).

So we can talk about the prevalence of systemic racism/sexism within our institutions and systems. I'm not going to deny those disparities. But let's delineate that discussion emphasis from the cultural and social aspects of it.

You claim there's a lack of "hatred" toward men. It seems to me that when people undervalue fatherhood, or minimize the effect of rape and harassment on boys/men, or egg on inappropriate male depictions in film/TV...anyone who would knowingly perpetuate those forms of marginalization are clearly EXTREMELY hateful. Even if they don't want to admit it to themselves. Even if you glibly refuse to hold them accountable for it. And just because you don't personally experience these things within a male body doesn't give you carte blanche to marginalize those of us who've endured (variations of) them.

And I contend that many men ARE indeed misandrist, themselves...even if they're not willing to admit it (and that unwillingness is probably largely due to their embrace of patriarchy, in the first place).