For example, the scene between Sonny and his wife is much more intense, than the arrival of Don Barzini who at that moment appears to be just another ordinary guest who happens to deserve the politeness of being welcomed personally.
I don't think Barzini was being portrayed as "just another ordinary guest." In fact, I think FFC went out of his way to show us that Barzini was a bigtime Mafia guy and potential rival. Barzini was the only guest, other than Johnny Fontane, that Vito was shown greeting personally. His title,
"Don Barzini," tells us loud and clear that he's a fellow Mafia Don. The studied reserve and coolness of their "embrace" is a harbinger of hostilities to come. And Barzini signals one of his guys to grab the photographer's camera and rip out the film--only a Mafia boss, not an ordinary guest, would have that concern.
One area of ambiguity at Connie's wedding concerns Hagen's status. Michael tells Kay that "he's a good lawyer--I think he's
going to be consigliere." [emphasis added.] But we never learn that he's actually become consigliere because FFC chose to delete an important scene, in which Vito, his sons and Johnny visit Consigliere Genco on his deathbed. To be fair, I didn't spend the entire movie (or any part of it) wondering what Tom's job was. But we don't learn that he was actually elevated to consigliere until late in the film, when Michael tells him, "You're not a wartime consigliere, Tom..."