The story I have read on that was Coppola thought he was going to be fired, in part because of his insistence on some of the roles, Brando in particular. Coppola signed his sister so that, as he is quoted "someone in the family would get something out of it".
The fact that he was way over budget did not help either.
The book was a best seller and Paramount was looking for a cheap "gangster movie" to make some bucks. Coppola had the vision to make it into two great movies and one disappointing one.
I love the first two Godfather movies but there are major credibility gaps in both. For instance, it always bugs me when Michael proposes that he be the one to kill Sollozzo and McCluskey and Sonny makes fun of him, saying it’s not like in the Army where you shoot a guy from a mile away, etc. “Nice college boy” and all that, and I’m thinking “Michael, according to the movie, is a Marine combat veteran of the Pacific island campaign, he’s seen more gory, up-close violent death than all his brothers and father put together.” It drives me nuts that nobody ever points this out.
That’s just one.