I suspect you're right that “no one with any prominence in this work" promotes the idea that humans are naturally racist. Alas, too many of their followers do. I'll add a PS to clarify that and mention that you brought it up in the comments.
However, I stand by the claim that class is what you have and identity is what you are. Gain wealth, and you're bourgeois. Lose wealth, and you're working class. This simple fact is confused by people who think class is about status rather than capital. I don't like the term "socio-economic class" because it blurs what should be distinct.
Speaking precisely, you are right that the poor are not a class—they're a subset of the working class. But in the vague American sense, they are: capitalists talk about the rich, the middle class, and the poor. Perhaps we need an update on Marx's lumpenproletariat to discuss the half of America that lives near or below the poverty line.
Which doesn't change the fact that Bruenig is right. The poor are not seeking respect. They're seeking an end to poverty, which the US could accomplish in a month with Basic Income.
So no, class is not about who you are. It's about where you're placed. Saying that being working class is your identity makes no more sense than saying being a slave is your identity. Your identity consists of things that do not change when your class does.
The socialist goal is not to turn everyone into members of the working class (though you could argue that was the goal of authoritarians who claimed Marx's legacy). The socialist goal is to end class so the idea of a working class is meaningless—the means of production should be controlled democratically.
Du Bois's idea of privilege made sense during Jim Crow and under South African apartheid—the law was different for people of different races. But social privilege and legal privilege are different things—a middle-class black person today gets far more social respect than "trailer trash" of any hue, and all the economic privilege besides.
We completely agree that race is false consciousness that must be confronted directly.
But your idea that white folks rather than the rich were responsible for sabotaging the New Deal, which had helped millions of black people and was responsible for black voters beginning to move from the Republican to the Democratic Party, is simply wrong. I will probably write a post within a week to explain why.
Till then!