The phrase itself is misleading, so it's something the right talks about far more than the left, even though it's affecting everyone.
So let's take seriously for a moment that it's actually happening, and society is changing in ways that make it unstable, and this is being done intentionally as part of a power grab. When people start labeling it as "cultural marxism" people on the left regard that as an silly overly-emotional McCarthy-era strawman argument. Which it is, in a way. Because the ideology that is being used to dissolve culture has changed, but the effect is still the same. Albeit marx intended this to happen in a good way, but it more often happens in a bad way, and those who want to grab power have made a clear note of this and use that accordingly. As it has been said, Marx was a great diagnostician to label the problem, but his ideas about how to fix it do not pan out in reality for various reasons, as all the various experiments of history have shown. It typically ends up being a consolidation tactic, so the government can be truly authoritarian as it wishes. And that's not to say that capitalism is immune from authoritarianism either, far from it.
Anyway, if this idea is instead reframed (more accurately) as "cultural takeover" or "cultural dissolution" or something like this, then we might be able to generate a discussion on this issue that both sides can relate to.
The origin of this cultural destruction relates to the same forces that helped destabilize society in Russia before and after the time of the 1917 Russian revolution. It got rid of the Tsars (who were legitimately bad rulers) and instead replaced them with Lenin, who used the ideology of Marxism/Communism (or at least his version of it, Leninism) to guide his rise to power in the hands of the public after the Tsars fell.
When any old order falls, and the new one arrives, those grappling for power will always sell us a bill of goods that looks great, but they never deliver. It's almost always a power grab. This is why some people hate "populism" despite it being basically the intended outcome of true democracy. Because it's often just a label that is used to deceive people. 3 million people unnecessarily died under Lenin, and 10 million under Stalin, who came right after and had one of the most authoritarian governments in history.
Lenin, and the rise of communism in Russia in the culture, was heavily backed by JP Morgan and the Rothschilds. This is a matter of public record, JP Morgan transferred $25 million to fund Lenin during his rise. Talk about foreign countries interfering in elections! "The people" didn't own the means of production, as in true socialism, instead the government owned it and vehemently shouted that it is "of the people, by the people, and for the people" but instead grabbed power and killed millions in the resulting starvation conditions created by the communist government that lasted for decades, and was very difficult to un-do.
Cultural invasion and dissolution is real. The label "cultural marxism" is a red herring designed to appeal to right-wingers and make left-wingers' brains shut down and refuse to entertain the argument, thus guaranteeing the topic itself is a "right wing" topic that can be split in a divide and conquer across the two parties. I think it is intentionally pushed in this manner, by ensuring "cultural marxism" is the only label under which this conversation can be had.
This prevents the public from unifying to stop the cultural takeover that is taking place. Instead, the larger the right-wing group that realizes this grows, the larger the left-wing opposition to this idea arises as well, thus keeping the first group in check. If you had to control 300 million people in regard to one idea, this is how you'd do it. Polarize the idea so heavily through the language used to label it, that only one side will actually see the intent of what is being said. And the other side is emotionally spring-loaded to reject that idea just because of the label and its associations, even though it's accurate in some ways but not in others. So the two sides will fight each other, even though both sides are being screwed. Because they're not actually arguing with each other, they're generally arguing against what they think the other person believes, based on loaded phrases like "cultural marxism" which immediately paints me like some idiot hand-wringing 1950s conservative.
I think it's important to avoid triggering labels like "cultural marxism" and instead use phrases that convey the same meaning (or are perhaps even more accurate) like "cultural takeover". This helps both sides see ideas clearly and not get clouded by pre-programmed emotional trigger responses, which will help us unify and thus actually have a chance to fix the problem.