I think you have a point that is key. Sessions “Approved” the decision to fire. The Decision “to fire” or the Recommendation to fire was by the work of OIG and taken to RR. The decision to fire was RR, and Sessions simply pulled out of the drawer an Approval stamp.
I remember hearing nothing from anyone but Rosenstein after the Recusal, and Sessions was working silently and could not get through hearings on the Hill, without help from his staffers. RR was keeping him in the dark, telling him Nothing, setting him up to look worse than incompetent, in front of Congress.