If we look only at the extent of the story, then I can believe this, but if we look at the reality, there is nothing like that what this woman said to get children, because I am also a Muslim and neither such a thing is in our religion nor has it been recognized by science.
It is a fact that some people face problems for children after marriage, but children can only be given by ALLAH Almighty. A girl or a boy can't have any children unless ALLAH wants to give, no matter what she or he wants, but we can change our destiny through prayer.
Whatever Aladdin and his wife are doing, they are doing it only to get children, like when a person is helpless, he is ready to sacrifice his most precious thing in exchange for what he wants to get. Just like that, both Aladdin and his wife are willing to do anything just to have a child. Therefore, it is also a saying that "children are a great blessing".
As far as the story is concerned, it seems to me that in return for the prayers of Aladdin and his wife and the good woman and whatever the good woman asked to do, Aladdin and his wife a virtuous child will be born.
But I would like to make one more point here as we have heard in proverbs that "to gain something you have to lose something" Will Allah Deen also have to lose something to fulfill his desire? Could it be his magic lamp? Or maybe something else that I don't know about yet, or maybe it's not what I'm thinking.
And yes, I really liked that woman when the princess asked to keep her in the palace then she could have enjoyed many of her desires by staying in the palace, but instead of fulfilling her desires by staying in the palace, she thought of going out and helping more people and in return for all that she did, no compensation or reward of any kind she demanded, I liked this good woman very much.
Now see that Aladdin has managed the egg by the time the woman comes back to the palace, and then what the woman orders to do....