Obito put all his faith in Kakashi, only to be shown by the world that it's not enough. It's not enough to care about your friends, they'll still die. It's not enough to give it your all, you'll still die.
His motivation is not simply "Rin died now I'm psycho," it's a fundamental breakdown of the optimistic ideals that kept him going. In some ways, it's not about her at all.
If you wanna hate on him because he "didn't cope like everyone else," fuck you. Honestly. I'm not saying he was right. I'm saying I understand. If you can't understand what it's like to have your entire reality collapse around you, and the rage, the hurt, the need to fix it- good for you. Claps all around.
Why the fuck should he care about Kakashi's suffering? They were hardly even friends. Kakashi helped make his life miserable and only showed up for him AFTER abandoning his friends. Dress it up how you like, but he did. Kakashi fucked up back then, and Obito isn't in a position to forgive. He was abandoned, left alone with Madara and Zetsu, having his mind poisoned by isolation and Madara's ideals.
But he should just walk away from that right? Oh poor Kakashi, suffering the consequences of his actions. Not poor Obito who died for him, gave his eye for him, trusted him, was abandoned by him, betrayed by him, isolated, brainwashed, and finally given one shred of hope by Madara alone and left to carry out heinous acts in the name of what he thought was a solution.
But fuck him I guess, cause he hurt your precious Kakashi and killed named characters instead of just some faceless guys.