I'm not arguing and I only came here because of my notifications, so I'm going to keep it simple. Misandry can be just as hurtful to men as misogyny is to women. If misandry wasn't a thing then I wouldn't have grown up to see the kill all men hashtag on both twitter and tumblr and there wouldn't be a book called "Boys Are Stupid! Throw Rocks At Them". There have been companies, such as Marvel, who have actually hired women over men and for some cases it backfired while in rare few others it didn't, but it's still not right. The best idea would be to hire the best person for the job.
Disney's ABC company has recently announced they are not interested in doing male lead shows. I personally think they could have worded that better because it does come across as anti-male. If the roles were reversed it would be anti-female and I think most people would be pissed. In March of this year, it has been found that Google is actually underpaying their male employees compared to women. Also do I need to really go into what happened in the comic book industry where there were comics books bashing men and pushing propaganda which is what lead to current decreased sales?
Basically what I'm trying to say here is men have feelings too, Phobo. I think the world would be a better place if people just stop putting each other down because of their gender. If there's something a man is better at than a woman then so be it. If there's something a woman is better at than a man then so be it. Both genders have their advantages and disadvantages.
BTW, I like the damsel in distress trope (it's a old, overused trope, but I like it). Plus, I really think you aren't looking at the right animes if you're bothered by women "playing second-fiddle to a male equivalent". I would suggest watching Kill la Kill, The Promised Neverland, Little Witch Academia, Ouran High School Host Club, Full Metal Alchemist (the women in this show are amazing), Michiko & Hatchin, and a lot of Ghibli movies. I think there's more, but I'm going by memory.
I don't agree with Sasukexbox's opinions, but I definitely don't think misandry is a myth after a lot of shit I've seen growing up. If you respond the way I think you're going to respond I'm going have to ask you to chill.