I don’t think the movie is horrible or anything, there are some good points, as such pointed out. It was, in many ways, supposed to be a big selling point of the movie- a warrior woman! Woo! I think though that part of the problem is even illustrated in the way you phrased this-
“despite China’s culture of generally looking to men for strength”
-I mean, there’s no distinction being made here (perhaps accidentally, but it is still lacking) between the Chinese culture of today and the Chinese culture of several hundred years ago, and they are very different, and their impact and perception of men and women are very different. Do kids also fail to make this distinction sometimes? Certainly: and it would only reinforce their ideas about a culture they know little of. I’m wary of Disney or any American film company presenting and trying to decode oppressive forces in other cultures for this reason.
And indeed, if I recall correctly, in the real Mulan story she worked for many years in the army and only unveiled herself later, and that was not really a main part of the story. So interesting in light of the above, perhaps, is Disney’s choice to make all the Chinese men aghast at her “deception” and deciding to kick her out of the army onto a snowy mountaintop, risking killing her.