I'm not convinced.

Although it's true that it's difficult to sustain a class struggle and a culture struggle at the same time, I don't think that there's much appetite for the culture war stuff among the wealthy. It's kind of hard to monetize, and it decreases stability, which is dangerous. Even given the danger of a class struggle, my read is that promoting a culture war is deeply irrational and mildly counterproductive for the wealthy (at any level--1%, 0.1%, 0.0001%, etc.).

Instead, I think the culture war arises from politics--from the part of politics which is not driven by money. Yes, you can buy a gazillion dollars of ads, but ultimately, you need to motivate people. Feeling that it's us-vs-them and the stakes are SUPER high is an incredibly powerful way to motivate people. The 1%, 0.01%, 0.00001%, etc. can get caught up in it too. They're also people.

And the culture war arises organically out of tribalism, now powerfully amplified by technology that allows us to indulge our preference for associating with those like us, and which monetizes our understandable fascination with extremes (they might be dangerous! or rewarding!) by exposing us to them constantly.

Income disparity is a big and real problem. But I don't see that it really has much of anything to do with the culture war, except by providing a bit of extra dissatisfaction that can boil over and/or elicit a bit more tolerance for extreme positions due to low investment in society. (But on the both the left and right, the wokest and supremecistest tend to be doing okay financially--you can't be struggling to put food on the table and have time for this stuff.) Other stuff gets left behind too because of this stuff--culture war issues have taken a lot of airtime away from climate change, from geopolitical issues, even from the later stages of Covid-19.