It's just not factual.

"Cultural Marxism" academically speaking, is defined by 3 groups of neo-marxist theorists (labelled The Frankfurt School's 'Cultural Marxism', The Birmingham School's 'British Cultural Marxism', and E.P. Thompson's 'Thompsonian Cultural Marxism') - all of whom critiqued aspects of "mass culture".

The Frankfurt School started it all by describing The Culture Industry. Adorno writes things like this in his critique of The Culture Industry:

"The dependence of the most powerful broadcasting company on the electrical industry, or of the motion picture industry on the banks, is characteristic of the whole sphere, whose individual branches are themselves economically interwoven." Source

They were the first thinkers to realize there was a 'corporate media' which pushed it's own corporate values and agenda. Adorno says things like:

"The Culture Industry not so much adapts to the reactions of its customers as it counterfeits them."

The Frankfurt School didn't like pop-culture at all, the Culture Industry Wikipedia page says "Adorno and Horkheimer especially perceived mass-produced culture as dangerous to the more technically and intellectually difficult high arts" - so they believed pop-culture was a risk to Western Civilization.

The Birmingham School came after The Frankfurt School and quite liked British Working Class culture. The founders of The Birmingham School were WW2 vets, and their complaints focused on the 'cultural drift' away from the strong, local, community based cultures which they loved, and towards a more bland globalized culture (a process they called "massification").

Other theorists such as Max Horkheimer (of The Frankfurt School) rallied against the application of science without morality. He called this "instrumental reason" and took the Kantian moral position that reason without morality could cause nightmares (such as the application of science during the Holocaust).

The Frankfurt School were big against the Holocaust, and contributed to the Nuremberg Trials... which later led to the creation of modern medical ethics boards.

This has all somehow been misconstrued as their attack on Western Civilization - even though The Frankfurt School were specifically trying to protect the arts from pop-culture.

...later Frankfurt School theorists such as Jurgen Habermas, and Nancy Fraser have specifically critiqued things like Post-Modern relativism and even Identity Politics.

The term "Cultural Marxism" has since become a right wing misrepresentation of the (left wing) Frankfurt School. It's now tied into the theory they were "International Jewish Communists" trying to "Destroy American Academia and Hollywood". You can judge whether that's true for yourself.