Of course Yinsen had included Kakashi in his calculations. Once you got on the bus, you still wanted to run? The doors had long since been welded shut!
Besides, Kakashi and Danzo had their own grudge. Even setting that aside, Kakashi had indeed gained quite a few benefits from the Uchiha Compound, and he did not mind giving something back. Moreover, the only two Uchiha left both had some connection to him.
Itachi Uchiha had been Kakashi's subordinate in the Anbu, and they had always gotten along well.
As for Sasuke Uchiha, there was even less need to mention it. He was Kakashi's student and, by Hidden Leaf Village tradition, could be considered his successor, the one who would inherit his will.
What was more, during this recent period, Itachi Uchiha had helped quite a bit so that Kakashi could learn how to use Susanoo.
Although Kakashi still could not truly use Susanoo, he already knew how it was supposed to be used. Next, he needed to raise his Chakra Levels, become more proficient with the Mangekyo Sharingan, and so on.
Fortunately, Yinsen was in no hurry. He continued leading the three of them all over the Land of Fire, occasionally crossing the border to sweep through nearby small countries. His goal was to find Orochimaru's lair.
"Why are you so obsessed with Orochimaru's hideouts?"
"Because Orochimaru has evidence of Danzo conspiring with him. Once we expose Shimura Danzo's collusion with Orochimaru, then no matter how excessive our actions are after we return, no one will sympathize with him."
Compared with Akatsuki or Hanzō of the Salamander, conspiring with Orochimaru was what would truly make the people of Hidden Leaf Village hate him, because Orochimaru had caused real harm to Hidden Leaf Village.
Unfortunately, Orochimaru had hidden himself too well.