First off, intersectional feminism only includes women, so TIMs should and will never be a factor because they are men, even if they believe they are not.

I think you might be misunderstanding what intersectionality is (and actually also confusing it with libfem values, which imo can be separate). I'm a black woman, and a bisexual radical feminist, my feminism needs to speak to the unique issues I have with race and sexuality that would be different because I am a woman, things that a black bisexual man would not encounter. Let me give you an example of intersectional thinking in feminism:

Case A: During the suffragette era, black women were made to march at the back of protests because white suffragettes felt it would make their message harder to absorb to the public. There were even racist suffragettes like Elizabeth Cady Stanton who wanted white women to vote because black men's votes were being considered - and thus wanted to strengthen the white vote and race. This is an example of race-exclusionary feminism. Thankfully we also had race inclusionary feminists like Sylvia Pankhurst.

Case B: During the rise of radical feminism, many radical femininists were still sadly homophobic - sidelining lesbian and bisexual women from the movement. The movement was failing to acknowledge the sexism of "stereotyping mannish man-hating lesbians" and distanced themselves from lesbians, they also had issues with addressing the other sexuality specific forms of misogyny such as corrective rape. A movement called The Lavender Menace: https://en.wikipedia.org/wiki/Lavender_Menace rose to confront this - they were very influential.

Case C: When it comes to gender stereotypes and the unique face of oppression each woman faces, sometimes race can alter how a woman is viewed. White women are stereotyped as chatty, feminine, demure, feminine. Black women are stereotyped as masculine, loud, aggressive, sexual. Asian women are stereotyped to be submissive, serving, quiet, exotic etc. These are all sexist stereotypes, and affect the groups differently.

Feminism gets called "white" when it only really tailors to the needs of white women. When white feminists banded up with women of color sometimes they would mess up - when challenging stereotypes that applied to them they would assume (often by ignorance) that what works for them must also work for others. For example, when white feminist or even black male authors made "positive" black female characters who were strong, independent, a bit aggressive/assertive, or sexual (things that white women fought to be seen as) - they did not realize they were furthering damage by solidifying a black stereotype that was harming black women from being seen as multi-faceted and even vulnerable human beings and aiding in their dehumanisation.

A lot of black women were attempting to find other modes of expressions where they could be seen as more rounded individuals, emotional, even feminine or demure if they wanted. A lot of them struggle with being seen as sex objects not worthy of a serious romantic relationship, a lot felt that even childhood and womanhood was taken from them very early on. The unique ways that colorism affected women more than men, etc. White feminists can not relate to such matters, thus a space needed to be made to talk about it.

Black women often felt that neither black men or white women understood that they felt confined to a small box of expressions - where white feminists embraced both masculinity and femininity in the cultural revolution, black women often felt they were stuck in the black strong female stereotype role which entrapped them in an epidemic of mental health issues. A lot of women of color in general felt they had to choose between their racial identity vs their womanhood which caused a lot of issues.

In short, intersectionality isn't, and shouldn't be (since there is a general cultural confusion even from intersectionalists) about calling this person a Karen, or culture wars. It's about acknowledging our differences, knowing how sexism has many forms based on culture and identity and acting on them. I know there's a lot of BS going around with white libfems calling other people "white" as an insult even with matters that don't even have to do with race, but that's just libfems being stupid with their identity politics bastardization. The concept of intersectionality is a good and positive one that all feminists should adopt and I am sorry they ruined the idea of it for you.