I’m inclined to agree with JarJarBricks here. While the movie might support that interpretation, the novel is pretty explicit that Kylo is still adamantly set on the past dying, and that he intends to usurp Palpatine after the Emperor has outlived his usefulness to him, creating a new rule with Rey.
As for the Eye recognising Kylo as the heir of Vader, I personally figure that’s only relevant as an answer to the question “Why would the Eye let him take the Wayfinder,” and I’m fine with leaving that somewhat ambiguous. The same way the Force is simply the will of the narrative in Star Wars, I like having the Eye as a mystical being who moves the plot along its path like similar characters did in old myths. That’s just me, though.