I think it's because the Emperor and Empress have a good relationship, and they also good parents, so the children didn't lack of affection so they wouldn't feel jealous of their new sibling.
My grandfather from my father side has 2 wives, he has 11 children from the first wives and 5 children from the second wives, they get along pretty well, even after my grandfather passed away no fighting over inheritance and such. The children of second wives often sleep over in the first wives homes and vice versa, and I never heard my uncle and aunt badmouthing their half sibling even behind their back.
maybe ive read too many webtoons about the royal family fighting but lol does it bother anybody that they haven't had an argument about the emperor suddenly having a child with a woman that's not the empress and it's all fine and dandy??? like no one's bringing it up? it's just completely okay?????? sounds kinda wack even if it's nice behavior