The movie, from what I could tell, is trying to demonstrate the ideas of how patriarchy works by getting the audience to feel bad for the Kens, who (like women in the real work) hold no positions of power and are the "second sex," accessories of the Barbies but not whole persons themselves. Then the Kens rebel, but they don't break the Ken/Babrie dichotomy, instead replacing matriarchy by patriarchy which obviously doesn't work. The main Barbie is clearly not OK with this status quo, hence why she leaves in the end, because she sees "her" Ken as a real person worthy of being enfranchised and elevated to the status of human. The Barbie world is just an inversion of our world, and it doesn't seem to me the movie is saying that's a "good" thing so much as it's using that inversion to explore what patriarchy is and what it means. Just because a movie shows you something doesn't mean it endorses it.
In no part of my post did I say that this movie presented the status quo of Barbieland's matriarchy as a positive thing.
One thing it fails to adequately address in its discussion of patriarchy is how it's enforced upon society. In the film's conception of patriarchy it's a component of society that has always existed and feminism (Keninism?) is a reaction to the natural status of things. As opposed to how the patriarchy of the real world is the product of centuries of hegemonic enforcement.
I don't think that the choice not to present the origins of patriarchy necessarily means it's being shown as the natural state of things. It just means that they didn't think that was an important thing to include. That's definitely a decision you can (and should) level valid criticism at, but I think that's different from saying they're presenting it as natural.
Ultimately, Barbie is essentially an extended toy advert with some basic level feminist thought tacked on. We can debate 'till the cows come home about what it did right and what it did wrong, but something so mass marketed will always be lacking when you try to get into the reeds of it all.
It does feel like the feminism is pretty core to the movie to me, perhaps even more so than the toy advertness (since the specifics of the movie don't really matter as much as that it's a hit movie named after and featuring the toy). However, it definitely does constrict the space within its feminism can operate, which I think is most clear in how it treats Mattel as well as capitalism in general.
Absolutely, I think you're right in that the movie is still fundamentally out of touch liberal feminism but perhaps I'm a bit kinder to it than you are. Certainly assuming that "natural" state of patriarchy is a major flaw with how all liberals think; nothing is ever historicised, nothing was ever created or evolved, it's always been like this and therefore to change anything is difficult, oft times inconceivable. To admit the patriarchy was made is to admit it can and will be unmade, and that it is propped up by centuries, millenia even, of violence, violence almost enitely absent in the movie. Granted, at the end of the day it's just a long ad for a toy anyway.