You lost me here. In reality, most southern and eastern European immigrants lived in self-segregated ghettoes (many of which remained for three generations). Most came solely for economic opportunity (not to "become American: whatever that means) and many initially came here with the idea of staying temporarily.
The attitude of the elite back then was, essentially, "we don't care as long as they don't overthrow the government or move into our areas." The attitude of the masses was negative, which is why the KKK had its peak years of membership from 1900-1925, when it was an explicity anti-immigrant organization.
The concept of the "melting pot" was actually an idea proposed by American liberals, as an alternative to the "cultural exclusionism" favored by the Right at that time. It never really was true, it took 3 generations for eastern/southern Europeans to fully assimilate (and become largely indistinct from other white Americans in terms of behavior and social mores), while blacks and Asians were not part of the "we" of the melting pot (even the liberals back then had their racial issues).
What we really need is enforcement of current immigration laws, an inculcation of constitutional principles among both the native born and the immigrants, as well as (dare I say?) a more inclusive sense of "we." That's my 2 pennies FWIW.
The part about "southern and eastern European immigrants" having resided in "self-segregated ghettoes" when they came to America 100 years ago is certainly true. And it is equally true that these autonomous communities existed for generations. (Many large cities, even today, have their own "Chinatowns." And "Little Havana" is, of course, a Cuban-American enclave in Miami.)
For quite awhile, this segregation was surely a reaction to bigotry by many native-born Americans. For instance, signs that bitingly declared, "Irish Need Not Apply"--or those with similar insults toward Italians or Poles--were once commonplace.
Fortunately, however, that mean-spiritedness is largely a thing of the past. And even those living in ethnically segregated communities today have assimilated to a reasonable degree.
I am less clear, however, as concerning your observation that "many [immigrants] initially came here with the idea of staying temporarily." What evidence do you have to support that view? (It has always been my impression that those who landed at Ellis Island had much more than temporary American status on their agendas.)
And the belief that late-nineteenth-century and early-twentieth-century immigrants did not especially wish to "become American" strikes me as odd. In order to gain their coveted American citizenship, they were required to renounce all other rulers and principalities--including those of their native lands--and, so far as I am aware, they did so eagerly.