Considering how low key Luciano and his faction were I'm sure they were just fine with the family going by "Genovese". Lucky probably hated his name attached to it in the first place. Wasn't his style.
I don't think that that is correct, Luciano was very flashy, as was Costello, they did nothing to hide who they were. They did a lot of hobnobbing with high society types, much more than Genovese. Genovese was the least flashy of the three, it just happened to be that he was on top when Valachi testified.
