Intersectionality & Marxism — Understanding Their Relationship

I've only recently begun reading theory and I've been discussing things with DeepSeek to check if I'm following the thoughts and arguments I'm reading. And let me add that I'm asking the questions below honestly, I'm trying to learn with the resources I've found and I'm trying to check the understanding I got from those resources (which includes AI slop) against the understanding of people who've actually read theory I haven't even glossed over yet.

Before I launch into backstory let me preview my question. Do modern-day leftists (hereafter referred to as lemmygrad) see Intersectionality as a counterrevolutionary development or as an advancement of Marxism?

• Backstory

  Today I was retracing one of the original threads I'd pulled to unravel the capitalist wool over my eyes. Years ago I had asked a middle school history teacher what CRT was and he said "it's Marxism, therefore it's nonsense!". I later learned that what he identified as "Marxism" was dialectics and my only knowledge of dialectics (this was before I was yet standing in the leftist light) was that it was part of the name of Dialectical Behavioral Therapy. DBT I knew of as a last gasp of hope for those with treatment resistant depression and, therefore, the association of Marx with the word "dialectical" had me deciding "maybe I'll read for myself some of what Marx was even talking about".

In my conversation with DeepSeek today about the lineage of Dialectics from Hegel and Absolute Idealism through Marx and Engels and eventually to Dialectical Behavioral Therapy. Things have been going swimmingly, learning a lot, but DeepSeek described a fundamental disagreement between Classical Marxism and Critical Race Theory (more specifically, with Intersectionality). I'm wanting to describe to you all (lemmygrad) the context and see if you think DeepSeek was oversimplifying, exaggerating, or simply hallucinating.

• End of Backstory

Back to my question: Does lemmygrad see Intersectionality as fundamentally counterrevolutionary to Marxism? It's my understanding that where Marxism is saying "no war but the class war", Intersectionality is instead saying "it's a bad idea to simplify it so far as that..."

A rough paraphrasing of the metaphor I tried out with DeepSeek (which I don't know is reflective of reality and not just shitty AI slop) is that in a scenario where the bakers (patriarchy/ruling class/dominant caste) control the bakery and are making shitty food for the proletariat. Intersectionality is saying "We need to take over the bakery and we need to figure out how to make good food." while Marxism is saying "the only thing we need to worry about or even should worry about right now is how to take over the bakery. We need to focus all energy on that objective and work on tomorrow's problems tomorrow."

Further clarifying Intersectionality's saying

if we don't know how to make food, and make good food that's fair for everyone, the system we make might be just as stacked as the bourgeoisie system, with the only difference being that it leans toward poor-white-men instead of rich-white-men but still away from women and minorities

I want to check what actual leftists (those who've actually read the theory and engaged in years of discussions with other leftists in real life) think about the accuracy of this simplification (that Intersectionality and Marxism disagree) and what do those in this community think about that disagreement?