I don't think that really captures the motivation or pressure to stop talking about class.
Rather, class in the US is really really messy (compared to, say, Britain, where despite the additional complications of social vs. economic class, you can describe yourself as 'lower upper middle class', and be understood), and the fact that speaking about class at all got you branded as communist (in the past), or socialist (currently), or elitist (always). America is largely branded as a classless society rather than trying to get to grips with class. Confusing the issue even more is the denigration on the right of "coastal elites" that are largely middle class.
As far as US political rhetoric is concerned, socialism seems to be the new communism.
Meanwhile, though speaking using the lens of racism isn't exactly easy (eg. "That's racist!" "No, you're the racist!"), it is at least somewhat possible.
Is it any wonder that politicians on the (US) left don't talk about class? Confusing your audience isn't exactly going to win elections. They can talk about creating jobs and inequality, but that's about it.
A bit sad to see what the author believes about his countrymen and he doesn't seem to have noticed that many of those already picked up on them while even having fewer prejudices.
My impression of them is that they were arguably mildly SocDem in the past, and then turned into (neo)liberals in the 90's. Liberals are not leftists.