There was no thought of making III when II was being filmed. Paramount worked on FFC for years to make III, until he finally agreed.
IMO, the real story of III is how Michael and Vincent used each other. If things had worked out the way Michael had wanted, Vincent would have continued to be the muscle that Michael needed to eliminate his perennial enemies. Vincent would then have been the head of a diminishing Corleone crime empire and Michael would have been totally legit.