My assumptions was right, but why would that old hag stole the child?

I think it was to gain sympathy with the princess, since they would have a missing child in common. And the old hag figured that if they have the princess’s kid, she’ll continue to feel connected through the trauma of a missing child and support them. With the daughter in their custody it also “guarantees”that she will never be found.

To bad yun tao isn't the type to be manipulated