I think the biggest problem with Godfather Part 3 is that it doesn't feel as informed as its forebears. For the first one in particular, Mario Puzo drew heavily from his sources in journalism, and a bit of his own upbringing as an Italian New Yorker with military service in World War 2. As a result, the film feels very real, drawing from actual accounts and experiences. Same goes for Part 2 to some degree. By the time Part 3 came around, Puzo had been out of journalism for nearly three decades, and it doesn't have the same feel of an inside track. By all accounts that film was made as a cash-in rather than because it was a story that needed to be told, and in my opinion you can really feel it.
Agreed with Hu that Part 2 is a bit overrated. It's more of a character study rather than the character journey of the first film, there's not much of an arc. Just deeper levels of exploration, plus the contrast with his father's relatively optimistic early years (which IIRC was originally planned for the first movie?). Also kind of a serpentine plot that can be tricky to follow.
Mafia stories can be fantastic, but it feels like there's one entry that absolutely dominates each period, and relegates everything else. Godfather did that for the Italian-American mafia golden age, leaving others like Once Upon A Time In America feeling a bit pointless despite their artistic merit. The Sopranos did the same for the post-government-crackdown period, retroactively curbstomping Goodfellas.
For a while it felt like there was no point in touching mafia stories after The Sopranos. We're in a new period now though, with globalisation changing the nature of organised crime. McMafia has tackled it and feels like something very fresh and new, but I think we'll see a really definitive take on the subject at some point in the next few years.
