You may possibly be correct. Or not. Doesn't really matter.
Revelation 20:7 shows that Satan is hurled into the lake of fire. When is death hurled into the lake of fire? Before or after Satan? If Revelation 20 is viewed in sequence, then death is hurled into the lake of fire after Satan is. (See Rev 20:14) If Satan is destroyed before death doesn't necessarily mean Christ is the one to kill him just because he's king. It makes sense that God would be the one to carry out that particular action regardless of who is king. Oh, and we must recognize that God is always king, even while Christ is. Just because Christ is king doesn't mean God is not. But, God has chosen to let Christ rule. That's what the covenant for a kingdom is about.
That said, who kills Satan does not affect our faith one iota, so it's not worth much of a debate.