Elwin wiped out the Yaxley family partly to establish his authority and let everyone witness the capabilities of the Cavendish family.
Another reason was that Elwin wanted to use the destruction of the Yaxley family to tell others that the Cavendish family had arrived, stepping into the magical world.
The purpose of annihilating this family was to officially push the Cavendish family forward into the magical world.
Chapter 231