I wrote this because while anime shows Emilia's promises for the nation, she never told anything about equality in the LN (only about fair selection). Actually, we don't know her or anyone's actual reason for doing it in the first place.
So, this is about their reason for the Royal Selection. For other 3(Priscilla, Anastasia, Felt), we don't know whether they have more than what seems yet.
Of course, below are all spoilers, so read at your own risk.
For Emilia, When her elven village in the forest was attacked and her aunt killed in front of her, 7-year old Emilia snapped and froze everything in the vicinity including herself, villagers and the forest. When she woke up from the cryosleep alone by Puck after very long time, she found Roswaal who claimed that Dragon's blood could defrost the people and forest. She believed it and choose to participate in the selection.
For Crusch, She had great respect for one of Lugnica's prince(who had one-sided crush on Crusch, which she never realized). When mysterious disease killed every royal members including the prince, Crusch felt deep anger that everyone cared only about continuing the covenant with the Dragon, without any actual respect toward the king or royal members. It is when she got chosen, and her reason for the promise, to cancel the covenant.
