The interplay between Santino, Jr., Vincent, Anthony Corleone, and Michael (Rizzi) Corleone would form the basis of the GF IV "coda" sequences.

I could see Santino Jr. getting drawn into the family business after suffering reversals in the "straight" world. He would obviously have jealousy/resentment issues toward Vincent, but might also see him as an ally.

Anthony Corleone, although he was antagonistic toward and rejected Michael in GF III, could be so shocked and disturbed by seeing Mary's murder that he might be obsessed with revenge (possibly spurred on by his mother?) - and see joining his father's organization as a way to get that revenge. (And it would also be a key plot line - who would he be trying to get revenge ON?)

And Michael Francis Rizzi Corleone, the baby baptized at the end of GF I, on the day "that all family business was settled," was even mentioned as "getting in trouble with the law" in GF II (in the boathouse scene where Connie asks for permission to marry Merle) - although that has to be a continuity error (he would have only been 5-6 years old at that time) - it's already established that Connie's children have a tendency to get involved in criminal activity.

Connie, I think, would have wanted to wipe away any memory of Carlo, who betrayed her father and set up the murder of the older brother whom she worshipped, so I think she would have dropped his last name Rizzi and officially changed his name to Michael Francis Corleone. And when he got to his 20s and 30s, after already being involved in criminal activities, it would be natural he would get into the family business.

So the conflicts would be already set up:

Vincent - Don Corleone, the anointed heir - at least to start out with, would have the backing of Michael and Connie, who had jointly assented to his takeover - although Vincent would always be suspect to the "legitimate" cousins;

Santino, Jr. - the true heir - who had rejected the family business and now wanted in;

Anthony, the 'pretty boy,' the honors law student/opera singer, who also rejected the family, but now wanted in for solely reasons of revenge (and who could be sure it might not be Vincent or even his own father he blamed for Mary's death?)

and Mikey, the traitor's son, who also would be angling for the support of his mother Connie, who of course also functions as sort of a Senior Consigliera to Michael and Vincent.

The whole thing could get very, very juicy.