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I think Asagi is also kind of a little warped.
For one, even if he almost doesn't have any friends, he's basically unable to go anywhere, ever, and he can never talk to people outside, unless permitted. At first one understands its for the contract, but later on, even when finding out the truth (and even after being told that he could break the contract and still have help for Aoi), he still chooses to stick around - I guess it's all in Kaname's predictions, seeing as I doubt he'd let him go at all. He's just giving Asagi the illusion of choice, to let him feel a little more free. That tidbit about him liking him, wanting to help him, and that he feels "caged" definitely contributed into winning Asagi over. But even then, wouldn't he at least have the contract broken, and just visit Kaname or date him normally? Also, not only does he never mention his best friend, he also seems to have decided to completely give up on ever talking to him again.
Seeing as how big his friend is supposed to be for him, you'd at least expect him to voice concerns about that guy, or ask for permission as according to contract. Not that Kaname would easily allow it, but I feel like if he got to learn the truth about his best friend, and who the true traitor is, as well as get to talk to him afterwards one last time, I think that would definitely provide a bit of closure for him.
And as far as I can see, the betrayal bit isn't completely cleared. Asagi never learned who the true culprit was, and Kaname didn't expand on it or properly clear his name. He just indirectly appealed to Asagi and said he wanted to start anew. Maybe the "clearance" of his name is simply that he completely denies being the traitor, barely any additional explanation needed. So maybe, in Kaname's mind, even if he did reveal who the true culprit is, Asagi might forgive his best friend and even decide to reconcile, thinking, "Ah, this guy just likes me, that's why", considering he seemed to accept Kaname's words like that. Or perhaps, it still lingers in Kaname's mind, that Asagi would choose to trust that guy over him. In that case, the current path in the story is the only one he thought is best - crush the other guy's hopes and spirit and guard the complete truth from Asagi, only letting him see pretty things. Because maybe, like before, he wouldn't be trusted.