In defense of Neil, Carlo break many rules when he took over. One of the reasons for the Appalachian meeting was to acknowledge Carlo as head of Anastasia's Family. It didn't happen then, a later meeting happened between Tommy Luchesse, Joe Bonanno, Jerry Catena and Richie Boiardo (for Genoveses) Carlo Gambino and Armand Rava. (some other members were also present but these were the main players). It was decided that Carlo would be a "temporary" Boss until the full Commission could meet once the heat from Appalachian calmed down. It was also agreed that nobody could get whacked without bringing it to the Commission.
So Carlo made his power moves, and by the 1962/63 meeting that was supposed to happen, Carlo removed all the major threats in the family. With the Valachi hearings, Joe Profaci's death/Colombo wars and beginnings of Joe Bonanno's problems in his family, Nobody could do anything about it.