the barbie movie is not "anti-man", it's anti-oppression.
in the real world, that oppression takes the shape of patriarchal power and women feeling like an afterthought or an accessory. in barbie land, it takes the shape of kens not knowing who they could become as independent beings because their existence has been irrevocably tied to barbie. barbie occupies the place of power, and ken is the afterthought and accessory.
the point of the movie is that any imbalance in the equality of any group of people makes the world a bad place to live in. ken feels unfulfilled and unappreciated in barbie land. He's been told his purpose is Barbie, but he's failing at that and doesn't understand what's wrong. He doesn't think he could be more than Barbie's love interest. Similarly, women in the real world feel forgotten, stunted, and held to impossible standards. Their purpose has been warped by other people telling them what it should be.
That's why it's So Important that both ken and barbie have their own reckonings of how they've reaped benefits at the expense of each other. neither of them wanted to hurt the other. Barbie just liked being a hero in Barbie Land and Ken liked feeling appreciated (and horses) when he was in the real world. But they both see and dislike how the other has been hurt by the power disparity in the real world and in barbie land. And they resolve to not perpetuate that cycle of hurt.
the reason barbie is a good movie is precisely because its main thesis is not Women Better Than Men. Nor is it a preservation of the binary or gender roles.
It's main thesis is that your identity is not the same thing as what you are to other people. And that's not exclusively a moral for Barbie herself. Sure, Barbie isn't Barbie because she's Ken's girlfriend - she's her own person to the point that the actively chooses humanity at the end. But a large portion of the movie also is devoted to explaining that Ken isn't Ken because he's Barbie's boyfriend. That Ken is his own person and should be allowed to be that, because that's (k)enough. Even Alan exists separate from the binary convention and has his own identity and story arc. He serves as a foil for the falsely symbiotic Ken/Barbie role dynamic.
anyway, my point is, no one should be (exclusively) defined by what they mean to someone else, or by what they have (whether that's power, a casa house, or a romantic partner). Everyone is a person deserving love, equality, and their own story, whatever that is.