In the original movie, we see Genco in a scene talking to Vito about the actress who is his girlfriend and later, backstage, we watch him quietly leave as the girl is threatened even though Vito seems more than willing to help him against the man in the white suit, the local mafia guy Fanucci whom he will coincidentally shoot later.
That is the major scene for Genco in this version but in both the book and the Saga, we see Vito and his sons visit the dying Genco who has become his consigliere.
Now it seems to me that the cowardice Genco showed backstage would be a factor weighing heavily against him as both a friend and key member of his gang although it could be argued that his cowardice did not mean he was not smart, even perhaps meant he was smart and therefore could be a good consigliere.
Could it be that the editing was because Coppola recognized that it would be inconsistent to have both those scenes in the same movie?
Is there anything definitive from the movie makers in this respect?
EDIT: I am not sure if I have this straight: The hospital scene was filmed for I and the play/backstage scene was filmed and shown in II. So the elimination of the hospital scene was not needed due to any conflict between the two scenes; Genco is mentioned in GF I only by Sonny as Paulie_D says. So the inconsistency, if there is one, between cowardice and being consigliere only occurs watching both movies together, whether the hospital scene is included or not. I know the hospital scene is in the book but in my opinion it does not work well in the film. (It makes the scenes with Vito being menaced in the hospital just more scenes in a hospital and maybe this, come to think of it, is why the Genco hospital scenes were eliminated.)