Same, I was thinking maybe the real daughter can't have children but then I realised that she did say in the first chapter that after she gives birth to her own child she will kill MC's baby.
So, maybe they thought the prince would have no interest in her. So they made the MC do all the work and once they realised that the Prince had a child with the MC they thought he had accepted her so MC and real daughter switched back places.
Lastly, maybe the Duke wanted to support multiple factions. Idk if there is more than one prince but he could have supported First prince and if that didn't work out he'll be like that's a fake here is my real daughter.
So, yh ur comment really made me question as to why they did this whole plan... hmmmmmm
So someone posted up a spoiler after reading I kinda understand.
http://www.mangago.zone/home/mangatopic/7814998/
It was mix between the first and last reason
She was incapable at first then she kinda wanted to let the other girl do all the work then steal it because she thought it would be fun
maybe she has already lost her virginity? it's the only opinion left
At first I thought it was because the real daughter was really in some way incapable to be there like actually on the verge of death so they had to use a fake but the real daughter said she was faking so it can’t be that
Second I thought it was because the real daughter did not like the king and didn’t want to have his baby but when they switched she seem like she was interested in the king and having his baby so that can’t be it
Third potential reason could be the fake daughter personality caught the kings eyes but I don’t think so be cause they schemed this way before how would they know that the king would fall for her or she would fall for the king, also it seem she said she has been taught various things by the blue haired lady for their schemes so they king didn’t like her just randomly
So why did the noble and his daughter use her and threw her away why doesn’t she seduce the king her self isn’t an unnecessary risk..... or did they do it all for fun