In a 2004 interview, Theodore W. Allen, who put the word "white" in quotes, explained why he shied away from the term --
"it's an abstract noun, it's an abstraction, it's an attribute of some people, it's not the role they play. And the white race is an actual objective thing. It's not anthropologic, it's a historically developed identity of European Americans and Anglo-Americans and so it has to be dealt with. It functions . . . in this history of ours and it has to be recognized as such . . . to slough it off under the heading of 'whiteness,' to me seems to get away from the basic white race identity problem."
For more on this topic see p. 78 n. 187 of “The Developing Conjuncture and Some Insights From Hubert Harrison and Theodore W. Allen on the Centrality of the Fight Against White Supremacy,” which is available in PDF format at the top left HERE
For more on Allen’s precision with words (including “whiteness”) see this video of a slide presentation/talk on Theodore W. Allen’s “The Invention of White Race”
HERE at minutes 36:49 to 39:10
For a Counterpunch article on "The Invention of the White Race" by Theodore W. Allen (2 vols., Verso Books), new expanded edition with internal study guides, Vol. 1: "Racial Oppression and Social Control" and Vol. 2: "The Origin of Racial Oppression in Anglo-America" CLICK HERE