The short answer: Because he's a joke.
The long answer: Kayn is not intended to be taken seriously. (At least, I sure hope not.) He checks off so many Anime Edgelord boxes it has to be intentional. And I'd argue it works. Unlike a lot of edgy types, no one respects Kayn or takes him seriously except his ninja dad and sometimes Rhaast. Akali thinks he's crazy, Aatrox ignores him in favor of talking to his scythe, and so on. By going that deep into the Arrogant Edgy Anime Guy cliche, Riot shows how hollow and pathetic such a character would really be.
Without his 'demon,' Kayn's a completely empty character. Compare baseline Kayn's dialogue with Rhaast and Shadow Assassin's. Baseline has a bunch of fun interactions between Kayn and Rhaast. Rhaast is genuinely funny, in an over-the-top violent sort of way. Shadow Assassin Kayn, though...he just goes even further into the Arrogant Edgy Anime Guy cliche. He's defeated an ancient evil, mastered his dark powers of darkness, and just generally has an ego and hairdo to rival Aurelion Sol. But there's no real depth aside from him sounding a tiny bit sad if he kills Zed in-game.
Kayn is ultimately not a Cool Guy. He's hollow.
EDIT: Oh, and the hair thing probably isn't a problem. You've got like a third of the roster with long hair that could easily get in their faces and they generally do okay.