It’s not really “killing” the EQ as it is just killing Regina and her soul. Her soul consists of the EQ persona and the Regina persona. However, with her ripping apart the EQ persona, it is wreaking havoc. But to make that persona go away, you have to kill the original or put it back into her. It makes total sense to me.
You know they could have easily shown this by having both men morph back into one after they die. This was my original thought process also but I can see RG point also. I would have loved the idea that EQ and Hyde were just manifestations of the original person’s soul, but to make that work they would not have any “real” power. They could cause chaos but not directly. Once did not show this they had the EQ control the Count’s heart, put up a protection spell at the town line, and maybe kill the Dragon. I don’t know I am a bit all over the place with this one