I'm not qualified enough to anwser your question, I just want to clarify something: you know that sentence is commonly tied to the fact that, when d'Azeglio said those words, "Italy" for many people was still a new and somewhat forced concept and thus they didn't really feel "Italians", right? So why do you feel the diaspora played a role in that?

As /u/bromao said above, Italy was in mere infancy when d'Azeglio said this. There was no Italian identity and most people, felt that the state of Piedmont in the north had essentially taken over, and as such felt no loyalty to this new state.

This feeling is often described as campanilismo, and if you have any more specific questions about the unification of Italy or Italy in the following years I'd be happy to answer, but I'm not sure if there is a way to answer your question as you have it phrased now

Alessandro Barbero made an interesting point in a conference, a few years ago (sorry, no written summary as far as I know). He pointed out that since the Middle Ages, Italy was always easily perceived from "outside", that is foreigners could easily spot Italians, and Italians abroad easily related to each other. However, when in Italy, Italians always felt different from one another. He quoted letters from members of patriotic associations, who actually dought for Italian independence, describing their travels in other areas of the country as if going "to Japan". In a nutshell, when at home differences always outweighed similarities.

This said, Italian diaspora certainly must have helped strengthening Italian identity abroad. But I am not really sure how much that influenced the Italians who stayed behind.