Well one route does make more sense. Killingham has signs of awakening visually drawn —> he is awakened unless Oda makes a special note as he’s done in the past.
he’s either awakened or another exception as Yamato is and the latter requires additional statements to prove than looking at his design
Since Luffy, Kaku and Lucci are the only confirmed awakened Zoan user, If at all, G5 wearing a white, instead of a black Hagomoro , He is the exception and therefore your argument bares even less substance.
However, G4 Luffy at Dressrosa already was wearing a Hagomoro prior to be awakened and so did Yamato, so there exists no etablished hard coded in-verse rule that confirmes it's automatically Awakening.
You simply want that route to be true, that's all there is to it.
Yes, I think looking at Killingham and saying “he can’t use awakening off the bat, so he’s not awakened despite looking visually like every other awakened zoan. Also only Luffy has white Hagoromo so he’s also not awakened” is not a genuine objective take.
this is Zoro fans and Aramaki’s sword all over
I'm not saying he can't, just providing evidence and logical reason why it doesn't make sense and yet I'm not disregarding the possibility entirely...
Black blades have been etablished and elaborated on at least when it comes to their distinctive design, it's pretty simple and straight forward, unlike with the Hagomoros.