I lurk the forums, having been a long time SKU fan from the 90s. I've been rewatching the show recently and came to these theories....

First, Anthy has magic on her own terms but it becomes coded as "evil witch powers" within the social norms.
If Dios/god and Akio/devil have power, then their sister Anthy must as well. I've always read this as just a family thing. There is a line that says "a girl who cannot become a princess is doomed to become a witch."  Anthy is excluded from the childish fairytale of being a princess/damsel in distress because she is the sister of the prince Dios. I'd argue that being the "witch" is more of a reference to her social function as a scapegoat and conceptual opposite to the pure "princess." This is more obvious in the film when Anthy is initially dressed as a princess but is called at as "witch" once she takes Dios away from serving the world. In short, her magic powers are unrelated to being a "witch" and more that her innate power was labeled as "witch" when society turned against her.

Secondly, Anthy largely suppresses her innate powers as Rose Bride.
The Rose Bride is a cynical take on the childish dream of being the princess searching for the prince on a white horse. Girls trap themselves in coffin-like barriers because they are waiting to be rescued by a prince and follow the "script" of what makes a good, proper life. This is simply the flipside to the "script" that forces other girls to become "witches" and scapegoats. As Rose Bride, Anthy is frustratingly passive and seemingly without any will because she is presumably waiting for an external actor (prince) to reassign her as a "princess" instead of a "witch." Her magic is limited to the game - producing swords and a projection of her body/Mamiya under Akio's direction. When Dios is called forth from the castle for the first time, everyone is surprised and it seems like it happened by accident. So in summary, the Rose Bride represses the extent of her powers to only function within Akio's game.   

Thirdly, Anthy is able to be free of her coffin and the Rose Bride games by embracing the alternative narrative.
This is why Utena did not "save" Anthy in the traditional sense - by bursting open the coffin door and lifting her out on a white horse before the castle collapsed. The show unambiguously shows Utena "failing" according to the script of the hero saving the princess: she is stabbed in the back, unable to reach Anthy, and seemingly dead in the ruins. Fairy tales do not exist. Instead, Utena saves Anthy by showing her a completely different kind of narrative option (friendship, same-sex love, strength in a little girl etc) and the bravery to pursue it. This is how Anthy is able to leave Akio and Ohtori as a whole. She is finally able to state her own will and desire to leave the coffin on her own terms. She doesn't need Utena or any prince to pull her out of it. This is also stated more obviously in the movie where Anthy is the one who DRIVES HERSELF out of Ohtori - she becomes the heroine in the final scene, not Utena.

OKAY THAT BEING SAID

I always wondered what Anthy did after crossing through the gates of Ohtori Academy. She says she's going to search for Utena in the real world, being neither princess nor witch but just herself. But....she still has magic, right? Now she is able to embrace her own innate powers and use them for her own goals.
Or is it that magic can only exist in a childish world that believes in stark binaries of Good/Evil, Princess/Witch? So leaving Ohtori, Anthy turns into a "normal" girl?