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I think it's popular because it's story telling is effective.
Much like in any other medium or culture, there is a noticeably large pool of works with an easily detectable formula with some isolated examples of sheer genius.
In the case of anime, I wholeheartedly agree that some VNs like Saya No Uta had incredible stories. But... I once compiled a list of hair/eye color combinations and the personalities commonly associated with them in anime. For every one insightful anime, there are twenty that take place in some school with a small army of batty females and a boring dude named Keisuke. In those cases, the story is monotonous and the plot is predictable.
I think anime is more popular given its excessive surrealism, which makes the mouths of many people in my generation water. I'm normally pulled aside to see something from Japan simply because it is weird or unpredictable, not because it is artistically "wholesome".