What "no emotional weight" means can most easily be seen by comparing The Irishman with The Godfather. Both are about Mob business and violence; however, who in The Irishman of the main characters can a viewer emotionally connect with? Russell Bufalino (Joe Pesci)? No, too much reptilian cold-bloodedness. Frank Sheeran (Robert De Niro)? He'd learned to kill in WWII, but many men came home from that war and, even if scarred, didn't become hitmen. Jimmy Hoffa (Al Pacino)? That character's myopic aggression and drive drops him from the dinner invitations list of both the Mob and audience. These three actors and their interplay are truly incredible, and it's a pleasure to watch their nuanced performances as the actors become their characters on screen. But do we care what happens to these characters? Not really, or if we do, it's more credit to the actors and audience than the characters' personalities.
In The Godfather, though, in a world of violence and murder, there exist Michael Corleone (Al Pacino) and his fiance and bride, Kay Adams (Diane Keaton). These innocents, and especially Michael Corleone, provide the readers with characters who can be identified with, innocents who are unwillingly sucked into the mire of the Mob. Even Vito Corleone (Marlon Brando) becomes more accessible because of the relationship between Vito and his son Michael.
In The Irishman, Sheeran's daughter Peggy is the most identifiable. As an adult, Peggy (Anna Paquin) says only seven words, yet she is the witness of the movie, the arbiter of right and wrong. She lacks the time in the movie, though, to carry the audience to its two and a half hour conclusion. At the end of The Irishman, we are left with no epiphany, no pot of gold, only the sense that we watched the lives of these characters unfold, and the only undeserved punishments received were, perhaps, those of the innocent audience watching mobsters get their deserved punishments.