Does this chart to the right, conveniently stolen from Kevin Drum really show that "class is still far more important than religion in America, despite the culture wars of the past couple of decades?" What it clearly does show is that class -- if, at least, by class you mean "income" rather than educational attainment or some other signifier -- has a real, large, independent influence on voting behavior. Across the board, as people get richer they get more Republican. This is true even of white evangelicals and of Jews, two religious groups oft said to simply leave their pocketbooks behind in the polling booth.
That said, the chart also seems to me to show that religious differences largely dominate income differentials. There are three religious groups -- Jews, the non-religious, and African-American Protestants -- such that the highest-income cohort of all three groups is less likely to vote Republican than are the lowest-income cohorts of the three other groups -- white evangelicals, white mainline protestants, and white catholics. And, again, rich white mainline protestants are less Republican than are lower-middle class evangelicals.
This, I think, is fundamentally what makes American politics tricky. It's often said that class doesn't matter -- or doesn't matter any more -- and our current politics is defined by culture. But that isn't true, and the chart clearly shows it. Movement up and down the economic ladder has a big impact on voting behavior. But religious affiliation also has a very large impact -- an impact so large it's doubtful to me that any sort of political strategy is really going to transcend it. Politicians and operatives just need to deal with a very complicated political landscape that's not amenable to the sort of simplifications that make for a good 800 word column.