I believe the one thing Kang didn't see, or saw it and wanted it to happen, was Loki not fighting and undoing Sylvie's actions in the past, but finding a way to live and thrive with it in the future, by letting all of the timelines having a way to live without being in the Loom.
In the final scenes of Loki and Kang talking, he goes through all possibilities, dialogues and actions he could have and can do with/against Sylvie to stop her from killing Kang, thus, restoring the Sacred Timeline. The "breakaway" from the time slipping and Kang's vision comes when Loki realizes there is nothing much he can do, and it'll always end the same; the Loom would explode and the TVA would be finished.
This realization leads him to time slip to the scene where he's being interrogated by Mobius; Loki knows what he has to do, cemented by Mobius' somber line -
"Most purpose is more burden than glory, and trust me, you never want to be the guy who avoids it, 'cause you can't live with the burden."
All of this finally leads to Loki going to the End of Time and replacing Kang, letting all the timelines thrive, at the cost of himself.