I've never seen this video. Doing anything "like a girl" I always thought it meant "like a little girl", which would imply, you're weak, because you're doing it "like a child". Yet, in the video, children (little girls) looked a lot more mature and strong than the grown women, who obviously grew up with the idea that girls are weak. It's sad, how it gives the impression that girls are strong when little, because they are themselves, but because of this stupid saying they grow up to be weak, because the world said so.
Do you not live in the UK or US then? Doing something 'like a girl' has always meant doing something badly, or stupidly or with no skill at all. The young girls in the video haven't yet been socialised to believe they are crap yet, or maybe the socialisation hasn't been internalised yet. That's why they seem happier in their own skins than the older girls or young women. The women (as they showed at the end) aren't weak at all, they have strong bodies, which can do amazing things, but they have been well schooled in society's belief that women are weak, and there to be pitied and ridiculed, when that belief is overruled, they realise just what they are capable of.
Women aren't pathetic wallflowers at all, we have all just been told for so long that that is what we are, that it becomes an internalised belief, no matter what the actual facts are.