It's more that the Democratic party is becoming the party solely of college-educated professionals who want to feel bad about social problems and the Republican party is the party of not-necessarily-highly-educated petit bourgeois who prefer to believe that everyone who is suffering deserves it (or the college educated people who react in opposition to the liberal culture war). Neither are working class parties, but the Republican pitch is easier to extend and expand to lower classes because it doesn't require learning special terminology or anything and because nobody who's already struggling wants to get told they should feel more guilty.

He's fundamentally pessimistic about Americans broadly relearning class consciousness enough for mass nationwide political organizing along those lines and so hypothesizes that mass organization will continue only to assemble based on cultural signifiers, and follows logically from that point. That's the main part that I think you can take or leave; I'm inclined to agree with his assessment that secondary education is one of the biggest cultural sorting mechanisms in this country.