Few years ago, when Rio was still trapped in the dungeon, and the whole world had announced him dead, - Amy was going through the toughest time.
Because she believed it was her fault he got trapped in the dungeon.
Her brother went into a battle without the support of his goddess who always looked after him, and it was all because of her stupid mistakes. (Hinted and talked about in ch160 & ch163)
After all, before the battle of the dungeon was going to take place, he had come to meet her once. Cause she messed up.
To get more experience about the world and be helpful to her brother, whose high talent and record breaking academic performance was putting him in danger every other day, with different talented heirs and students ganging up against him_ Amelia chose to join the adventurer's guild in secret.