The original Kakashi had been deeply depressed after losing his family, friends, and teacher one after another. If not for Might Guy and Jiraiya's books keeping him company, he probably would have broken down long ago.
But geniuses were beings who broke convention. In his youth, he had nearly reached heights that ordinary ninja could spend their entire lives striving for. Thus, even after sinking into despair for so long, he still stood above most people in the Ninja World.
After learning that all his tragedies had been deliberately caused by others, he regained his drive. He was no longer muddleheaded and had broken free from his former constraints.
Regarding the Uchiha Clan and the village's higher-ups, he no longer blindly trusted the Hokage's authority. He saw the true nature of things and chose to stand with the Uchiha.
As for Obito Uchiha, he no longer blindly believed that Obito had some hidden hardships. Uchiha Madara's manipulation was one factor, but murdering his teacher and teacher's wife, destroying his own clan—those acts meant he had utterly abandoned the most basic principles of being human. This was not something that could be forgiven merely by bringing him back to atone. Only by denying his existence could Kakashi give his former best friend the best possible answer.
Having escaped his former confusion, he also began consciously improving his strength. As a result, over the following days of sparring, Minagawa Uchiha found the fights increasingly difficult. At the same time, he gained more and more from them.
Kakashi had always been a genius. His talent even surpassed that of Uchiha Itachi, a genius who could perform the Great Fireball skillfully after seeing it once. At barely over ten years old, Kakashi had even created the Chidori, a technique Second Pillar would rely on