I think Hagen was out because Michael held him responsible for Sonny's assassination. In the novel, Hagen beats himself up badly after he learns Sonny's dead. If I remember the text correctly, "Hagen knew he wasn't a wartime consigliere...Genco would have smelled a rat."
In both the book and the movie, Michael tells Hagen he's out with the Don and both Capos present. This could be interpreted as Michael's judgment on Tom for Sonny's death. Neri's in that scene. Towad the end of Part II, when Michael asks if Tom is going to leave the family and become "vice president of the House and Hotels," Neri gives a sly look, indicating that he ratted out Hagen, presumably to become consigliere