The first question is to determine what Nick's failures in sensibility are. Nick is a reliable narrator and does his best to live up to his own expectations of himself in being honest, objective, and one who doesn't judge others:
Reserving judgments is a matter of infinite hope. I am still a little afraid of missing something if I forget that, as my father snobbishly suggested, and I snobbishly repeat, a sense of the fundamental decencies is parcelled out unequally at birth.
Nick does seem to be a good judge of character and he does reserve judgments until he's gotten to know someone. If he fails (in sensibility) in being unable to respond emotionally, it might be that he fails to commit to a relationship. He evidently left a relationship when he moved to the East and he doesn't commit to Jordan.
There is a tendency to identify Nick's failures in sensibility with Fitzgerald's but the reader could just as well separate Fitzgerald from the novel. This depends upon the reader's interpretation. However, if the reader is seeking a connection between Nick and the author, (s)he could attribute Nick's development from romantic to cynic as an expression of Fitzgerald's cynical illustration of the death of the American Dream. The novel presents elements of romanticism and idealism (mostly with Gatsby's reason for being) but those hopeful qualities are not absolutely pure and in the end they lead to failure. In that respect, Nick's inability or refusal to commit or respond emotionally is an expression of Fitzgerald's more general cynicism about romanticism, love, and the American dream. Also, consider how proud Nick is of his own honesty. If we interpret this as Fitzgerald's honesty as well, the point could be made that one who knows or believes he is more honest than other people in his life is less likely to get emotionally involved with others because he is already thinking that everyone else is necessarily less honest than he is. In other words, Nick finds it hard to trust another person because he doesn't think they will be as honest as he is. And this could be a manifestation of Fitzgerald's thoughts in his own life.
Nick does, however, respond emotionally to Gatsby, but only just before and following his death. This is why Nick is so outraged when no one attends Gatsby's funeral.