She hade to hide it because the competition is only for men. A lot of people know about how skillful she is in swordsmanship, in the beginning of the story the knights were jealous of her. They were jealous because she was better then ALL of them.
It'll repeat what happened in her last life. She doesn't want to be recognize anymore that's why she doesn't care about her family anymore and simply want revenge to the royal family. Going back from the first chapters, she was forced to be a sword because she flaunt her skills and abilities and recognized by the king.
No it would be bad if they knew who she was because one the emperor might take her away from the 13th prince thinking that she could be useful for some other purpose like being turned into a sword and right now she wants to stay hidden and control the 13th prince so revealing her identity will prevent that
Ik why she decided to hide her identity but I still feel like it’ll be really cool if everyone knew who she was in the swordsmanship competition. That way her family would know they missed out on a gem that’s even better than her brothers and for everyone to stop belittling her as just a knight cause of her family title