As the leader of the top ten families, the Wrights of Wolsing had their own duties and responsibilities. They needed to maintain the high and mighty image of the top families in the hearts of the commoners.
Once this was fully exposed, perhaps the people from the precinct and Dragon Cell wouldn't take any action against the Wright family, but Emery might personally eliminate him in the name of justice to protect the family's reputation.
Since Emery hadn't truly become the leader of the family