In many respects, part of the reason why the family is depicted in such a bizarrely warm manner might be the reflection of the time period in American Cinema. The 1970s was a period in which independent filmmakers sought to transform how individuals see themselves, their world, their art, and traditional conceptions into something new and innovative. Coppola's rendering of a mafia boss as a sympathetic and, almost benevolent, figure might be reflective of this. In contrast to the traditionalist monolithic view of the gangster as presented in Jimmy Cagney films and Howard Hughes' "Scarface," Coppola rendered his gangster in mythic sensibility, allowing the audience to see the mob in a different light. Don Corleone and his entire family is not presented as participants in organized crime as much as a King with three sons and a kingdom to control.
Everyone in the Godfather sagas are alazons (imposters, criminals, bad guys). It's just that the viewer only has access to the Corleone family, and not the police or detectives trying to catch them; as such, there are no real heroes. In fact, even the cops are bad. So, we view the Corleone anti-heroes as the good guys, or the lesser of the other evils. Compared to the other alazons: the Tattaglio family, the crooked Capt. McCluskey, "The Turk" Sollozza (the last two Michael executes), we see the Corleones as the best of the worst.
As the Enotes editor points out (from The Godfather novel):
It occurs in the sentence, "We will manage our world for ourselves because it is our world, cosa nostra." In the same speech, he says, "We are all men who have refused to be fools, who have refused to be puppets dancing on a string pulled by the men on high."
It is a noble sentiment, and certainly one that gains more sympathy from the reader than would be gotten if Don Corleone said that he ran his criminal empire to make money.
In comparison, they repeat another phrase throughout The Godfather trilogy that echoes in our ears: "It's not personal; it's business." As consumers and capitalists, we condone their atrocities because their business rhetoric appeals to us. Murder, extortion, and racketeering are just part of the business, they say, and we believe them.
Also, the Corleone family honor seems stronger than the Tattaglia's, who want to go into the drug business with Sollozzo. Again, we view the Corleone's sins of controlling the gambling and booze rackets as forgivable in comparison with the selling of drugs that Sollozzo advocates.
Although Don Corleone is the fictional leader of one of the five families of the Sicilian Mafia in America, like many real-life godfathers, he is proud of his Italian heritage and considers honor as the most noble trait of them all. Unlike the other families, the Corleones attempt to use murder as a last resort. There is always the talk and hope that the family will one day be "completely legitimate." Young Michael Corleone has returned home from World War II as a hero (the movie never states how he earned this honor), and Don Corleone hopes to keep him out of the family's illegal operations while grooming him for politics. (Much of this is based on the history of the Kennedy family. Bootlegger Joe Kennedy always hoped to see his children in the forefront of politics; his success was much more apparent than Don Corleone's would prove to be.) The word "family" is sacred and close-knit; Mafia members' relatives as well as the non-related gang members show unity and revel in their privacy. Don Corleone's "warm-heartedness" stems in part from his code that a favor is later rewarded by a requested favor; so, while he often gives freely to others, he expects to later be compensated to an even larger degree. Many mobsters have seemed friendly and out-going to the public (John Gotti, Frank Costello), but their ruthlessness is usually evident.