Nagato and Obito are not that similar if you think about it. Nagato just went on this angry rampage without a plan, while Obito got brainwashed by Madara that the world simply sucks. And Obito additionally convinced himself that the world is worthless since he’ll make a new one, and every sacrifice and horrible thing he did in it didn’t matter. Apart from when Rin died, he never ever killed out of anger.
Made a rational decision that the world is horrible and put in hard work to change it.
Not caring about Kakashi showed how convinced he was. If he had reacted to him in anger, that would be a boring manga trope