Something like that. There's also the fact that the magic tower and imperial gov are at odds with each other, as the imperial family's divine power and the mages' magic are each other's natural enemy.
She doesn't care about the emperor, she only decided to stay because if she left now it would feel like unfinished business. She grew up seeing Elincia as a perfect girl and the perfect daughter because of the book, and seeing her not only behave weirdly but also not getting along with the emperor is confusing Philomel because it goes against everything she believes in.
Remember, Philomel is often questioning what "family is", because her trauma revolves around that. She was deeply convinced that she does not deserve love because she is not the emperor biological daughter, while Elincia is.. Elincia is a very special being to her, so yeah she needs to investigate why shes different from the book, her own life's meaning depends on it.
Also, her actual father is basically the anti - thesis of the emperor, he is like his natural enemy, so telling the emperor that she is the daughter of the master of the tower (who is know for being chaotic) would only bring her trouble especially since it also means that she was born through an experiment like her brothers, who have a really bad rep because of it.
Someone remind pls why Philomel hasn't said anything about her bio family to the Emperor? I kinda forgot or maybe I missed the details. Why didn't she just say straight up that she's found her real family? Does she know about the fake Elencia already and decides to stay in the palace to investigate?