The relevant scholarly literature seems to have started with Noel Ignatiev’s book “How the Irish Became White,” and taken off from there. But what the relevant authors mean by white is ahistorical. They are referring to a stylized, sociological or anthropological understanding of “whiteness,” which means either “fully socially accepted as the equals of Americans of Anglo-Saxon and Germanic stock,” or, in the more politicized version, “an accepted part of the dominant ruling class in the United States.”Some of my graduate research in history involved looking into Irish immigration to the US from the 1830s to 1850s and the nativist response to it. At the time, the Irish were considered white even by the Anglo-Saxon Americans who opposed them. Race was understood quite differently then than it is now (as Bernstein points out later in the article), and the idea that the Irish were not white when they arrived uses today's race and ethnic studies definitions and projects them onto American society in the past. It has nothing to do with how Americans in the 19th century viewed the Irish and everything to do with how race and ethnic studies researchers view race today.
Those may be interesting sociological and anthropological angles to pursue, but it has nothing to do with whether the relevant groups were considered to be white.
Here are some objective tests as to whether a group was historically considered “white” in the United States: Were members of the group allowed to go to “whites-only” schools in the South, or otherwise partake of the advantages that accrued to whites under Jim Crow? Were they ever segregated in schools by law, anywhere in the United States, such that “whites” went to one school, and the group in question was relegated to another? When laws banned interracial marriage in many states (not just in the South), if a white Anglo-Saxon wanted to marry a member of the group, would that have been against the law? Some labor unions restricted their membership to whites. Did such unions exclude members of the group in question? Were members of the group ever entirely excluded from being able to immigrate to the United States, or face special bans or restrictions in becoming citizens?
If you use such objective tests, you find that Irish, Jews, Italians and other white ethnics were indeed considered white by law and by custom (as in the case of labor unions).
One factor that Bernstein does not touch on, probably because it is not widely recognized, is that nativism in America prior to the Civil War was not about immigration per se but rather religion. The nativists had no problem with many other immigrant groups coming in, but they had huge objections to Catholic immigration.
Read nativist writings and over and over you will read about how Catholics can never be true Americans because they owe their final allegiance to the Pope, whom nativists often depicted as a foreign prince. As millions of Catholics poured into the US, they developed a separate Catholic school system, avoiding one of the main ways immigrants were assimilated in the North. There were legal battles fought over whether states could mandate that schools use the Protestant version of the Bible (it wasn't questioned that they could mandate study of the Bible). The separate school system and various other Catholic social organizations that sprang up seemed like an effort by the whole population of Catholic immigrants to avoid becoming American. That's why the "nativists" (an epithet invented by their political enemies) called themselves "native Americans" and formed the American Party. As Catholics became important voting blocks in Northern cities and began to exercise political power, the nativists began to view mass Catholic immigration as an invasion by a foreign power.
All of this built on centuries of anti-Catholic sentiment in England which came to America early on. The Puritans, after all, wanted to purify the Church of England of all its Catholic aspects. Anti-Catholic bigotry was probably the oldest kind of bigotry in the American colonies, and it continued into the new nation. I've read that at the Constitutional Convention there was a debate over whether Catholics should be allowed to vote. The winning argument was that there were so few of them in the nation that it couldn't hurt anything to let them vote. That began to change with Irish immigration during the famines.