Cuz he is a treat to her husband, to her future children, for the political power of the royal family and the emperor loves his brother too much to check on him so she has to, and is not like she hates him either she seems to be in quite friendly terms with him if she bantly says her mind and let him trample with the palace jewelry per example
And ML may not want the throne but he being a member of the royal family is enough a treat cuz other people can use his existence as a tool, cuz his children have a claim, cuz no one knows the future and the people can start wishing for the duke to take over instead of having a weak/sick/old/childless/not a warrior/young ruler, there will aways be a comparison and "option" to just trow her, her husband and children out of the way and how and why they chose to do that is not in her control
What's wrong with the empress? Why is she so hostile? The duke doesn't even want the throne!? Since the emperor was so nice I thought she would be too, but why is she like that! ಠ︵ಠ