Well what I mean when I say “of his own” I mean that it doesn’t have to be in service of another character, which is what you seem to be suggesting it has to be.
I’m suggesting it has to be because the story is REY’S story, not his. He only starts to creep into co-lead status 2/3rds into The Last Jedi. What you’re effectively asking is “why isn’t it Kylo’s story” and my answer is “Because it’s Rey’s.” - and it is. It’s not both of theirs. It’s hers. The “two sides of the same coin” thing got a lot of play, but her side of that coin HAS to be the one that lands face-side up at the end because it’s HER story. And really, the “Two sides of the same coin” comparison only really works as a reflection on, and reference to, Rey as the main character. She’s the focus.
The decision was made to make it her story for TFA, and that decision was underlined in TLJ. Abrams never put in the work, much less did that work well enough, for there to be a solid case that by the third movie, Kylo’s name should sit right next to hers. It COULD HAVE. But I don’t think the case was ever effectively made, and if a lot of that case rests upon the poorly argued precedent that Lucas himself set by trying to awkwardly (and ineffectively) retcon “the meaning of Star Wars” to be ABOUT Anakin’s redemption, then that’s not a good case in and of itself, and that precedent wouldn’t do Kylo’s redemption any favors. And it didn’t.
I’ve never said redemption was impossible. My primary beef is with the notion that redemption was the ONLY possibility, and therefore storyarcs that would have led to Kylo’s character staying firmly villainous were abandoned out of hand. I’m not arguing against redemption so much as I am arguing for the premature and short-sighted nullifying of story possibilities for the sake of adhering to broken conventional wisdom about what “Star Wars really is.”
If they were going to redeem him, it needed to be better than this, more well thought out, and more thematically rich - and it needed to be a key part of REY’S story. They didn’t really do any of that. And if they could have served Rey’s story better without driving towards redemption, but they decided against that for no other reason than “That’s not Star Wars” I feel they shot themselves in the foot a little.