It's been said that at various times, FFC and Puzo considered script treatments that would have had Anthony in the business, but evidently they rejected that approach as lacking credibility. Sonny's sons might have been successors, but they were essentially unknowns. I suppose they chose the Vincent character because he was illegitimate, but a family member nonetheless. Hence, give a break to someone to whom the family has an "obligation" and perpetuate the family, as in Michael saying, "Nephew, call yourself Vincent Corleone," and Vincent kisses his hand. and, as a favorite of Connie's, she got a more assertive role.
The romance of cousins has no tieback to the opera, "Cavalleria Rusticana." What does tie in is the theme of vengeance in Sicily. Also, note that Vincent threw down his challenge to Zasa early in the film by biting his ear--as does the "Cavalleria" character Turriddu to the character Alfio (we see this in the film, and Vincent grins knowingly).