Interesting thought, Olivant! I don't think this topic has ever been posted before--will wonders never cease?!!
I come at it a bit differently. I've always believed that the GF films are NOT about the Mafia per se--it's a Trilogy about an Italian-American family, with the Mafia as a defining milieu. (In the same way, "Gone With the Wind" isn't a Civil War film, it's a story about a family with the Civil War as the defining milieu.)
With that in mind, a very good film about an Italian-American family could be made against today's Mob background. We'd see a parallel set of pressures on members of the family, updated: to get involved with drugs or not; the constraints of living with updated and improved electronic surveillance, the constant threat of RICO and betrayal by plea-bargainers in the family, competition from other ethnic gangs. I also think a modern GF could derive considerable tension from the fact that opportunities to go to college or otherwise find success in the straight life would be constantly tugging at family members.
Some excellent attempts have been made along this vein--"Mean Streets" and "The Brotherhood" come to mind. But they weren't focused on a family in the way that GF was. "The Last Don" was a move in that direction, but not a very good one in part because it didn't really emphasize the changing milieu in the way you're suggesting (and it was crammed with all that Hollywood crap that Puzo felt compelled to show off). Maybe someone else will do better.