Not one of the other New York Families were ever going to let it happen.

The Genoveses were being run by Funzi and Benny Squint, The Lucheses, who were Carlo's biggest allies when Tommy Brown was alive, due to family alliances, were being led by Carmine Tramunti, while Tony Ducks was in prison, and had strong links with the Genoveses. The Columbos had Persico as Boss, and he would never kowtow to Gambino and the Bonnanos had Evola and later Rastelli. Rastelli was a weak Boss, and while he may have not had the strength to stand against Gambino himself, other members of that Family would never have let it be taken over.

Carlo probably had the biggest influence of the Commission members in the time under discussion, but I think his actual power had lessened since Tommy Luchese died. I also believe the Genoveses were probably the strongest Family again by that point; assuming they were ever not.