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). Indeed, some lighter-skinned African Americans of mixed heritage ”passed" as white by claiming they were of Arab descent and that explained their relative swarthiness, showing that Arab Americans, another group whose ”whiteness" has been questioned, were considered white. By contrast, persons of African, Asian, Mexican and Native American descent faced various degrees of exclusion from public schools and labor unions, bans on marriage and direct restrictions on immigration and citizenship.
You can also get a sense of who was thought to be white by considering whether Americans considered a particular marriage to be an interracial marriage; only 4 percent of Americans approved of interracial marriage as late as 1958. Yet Anglo-American whites were not ostracized by polite society for marrying Irish Americans or Italian Americans. Famous Jewish Hollywood stars such as George Burns not only married Gentiles, but openly partnered with them in their careers. We know that light-skinned Cubans were considered white at least as of 1950 because (despite the trepidations of the studio) the public accepted Lucy and Ricky, in a way they would never have accepted a black-white or Chinese-white couple. American Indians were considered non-white, but if they assimilated and married whites their children were generally accepted as part of white society. Did you know that Will Rogers was 9/32 Cherokee?
When I've pointed this out to people, they often rejoin that people in the late 19th and early 20th centuries often referred to the ”Irish race," the ”Italian race," the ”Jewish race." That's true, but they also referred to the ”Anglo-Saxon race," and the ”Teutonic race," the latter two generally considered to be superior. The racist pseudo-science of the day divided Europeans into various races by nationality or perceived nationality, and often created a hierarchy among those groups. But that was a racist hierarchy within the white group, not evidence that these groups weren't considered to be white. This point is often obscured by the whiteness studies crowd, because racism within a white hierarchy conflicts with their understanding of American racism solely being about ”whiteness."