My family is from Depression-era Jersey City and while there were certainly ethnic clubs (including the infamous German-American Bund), my familiy did in fact know people whose parents so wanted them to be American that they'd get hit at home if they spoke their parents' language at home instead of English. And my grandfather did sit my two Scotland-born uncles down, when my grandmother finally joined him in the US with my uncles, and tell them that they were American know and if they wanted to be Scottish, they could go back to Scotland. You are correct that there were people who lived in ethnic ghettos but there were plenty more who realized that their ethnicity was contributing to their problems who wanted to be Americans.
The Jersey City of my grandparents was an Italian Ghetto where one did not need to know English if they didn't want to. My grandparents did not know English until the went to public school and had to act as interpreters for their parents. The same was true of my father's family in Newark, who attended Polish language parochial schools, but learned English out on the street or through movies.
We must be careful before we idealize the past. It was World War II that assimilated the European ethnics, and made them realize they were Americans. Then came the red scare of post-WWII in which to be "foreign" was to be suspect. The "everyone wants to be American" concept is largely a product of postwar America.
In any event, it doesn't matter, every child of Colombian, Cuban, Cambodian, and Korean immigrants I have met speaks English better than their parent's language.