It's not about who killed the night king, it's about when he died. If the opening scene of the show is about the white walkers coming and killing everyone, that should be the final scene of the show. Imagine return of the Jedi beginning with killing off Darth Vader, the main villain that the protagonists have been fighting against the entire time, and then the rest of the movie is Luke trying to defeat Jabba. Or if frodo got the ring to Mt Doom and then they still had to defeat saruman

They should have lost the battle in winterfell, retreated to kings landing, then had a final battle against the night King and cersei all in one major battle. It truly doesn't matter who kills the night King, just that he is the last antagonist to die since he was the first one to be introduced. It's about closing plot threads in the order that you open them.