I think you're completely off-target here because you've conflated "argue my ideas" with "engage in a social media popularity contest".
This is not how one argues ideas.
The entire thing about Raya vs. Airbender is exactly about people switching away from arguing about ideas.
Indeed, these things almost always turn quickly and viciously into referendums on the character of the person who transgressed--the idea might be the transgression, but the person is the target.
Also, A.P. said "argue my ideas, not my identity" but didn't insist that every idea be dissociated from every other idea. This allows a reputation for ideas-worth-hearing to be built up (or destroyed), and that's fine, certainly as long as the reputation is only being used to direct attention.
Also, while I appreciate your comments about intersectionality being potentially useful, I think the framing of "intersectionality" is at the very least woefully under-proven and probably just outright damaging, especially when coupled with other features that tend to come along for the ride with intersectionality (like the power of narrative, "speak your truth", "lived experience", etc.). It's stereotyping on steroids, with an ample helping of postmodern-style non-commensurate and therefore mutually incomprehensible experiences.
Seriously--what are we supposed to do, most of the time, other than stereotype others with this information? "That's just the black lesbian southerner in you talking!"
If the idea of intersectionality helps to make people sensitive to others' backgrounds, good on it! However, I think we should find a different, better way. The reason is that all the dividing into categories accentuates our difference and encourages tribal reactions: I won't react to this as me, I will react to this as a neurodivergent Italian. It tends to deliver pre-formed perspectives to people that they ought not question or else they lose their group membership (which prior to intersectionality they wouldn't have thought they had, nor would nearly so many people have judged them on it: "that's not neurodivergent-Italian of you" tends not to occur unless you're constantly doing the intersectional thing).
Instead, we should conceptualize it as a blending of different backgrounds, not a slicing and dicing into boxes. You might be Pakistani, male, a middle child, a parent of twins, a Dodgers fan. You don't go, "Oh, how would a Pakistani middle child with twins handle this situation?" That's ridiculous. You blend them. Like A.P. said--it's holistic. People are, by default, holistic. (But they switch easily to tribal identity when prompted.)
The reason why intersectionality is useful in law is because using it is bad. "Ha ha!" cackles the villain. "I will not discriminate against blondes. I will intersect with height and only discriminate against tall blondes, muahahahaha!" It is precisely because people apply prejudice along intersectional lines that we need to be aware of it in a legal setting.
And sometimes in a social setting too, because people do sometimes engage in tribe-specific actions, and within the context of that tribe they may face hostility or different circumstances because of other qualities that are shared with some subset of the tribe. Fair enough--in the cases where the pattern arises naturally, and is sufficiently bad to demand attention, notice it, attend to it, try to fix it. (Or notice it, realize that attending to the differences is causing the problem, and encourage people to focus on other stuff.)
Addendum: needing to view things with intersectionality can come up because we already were viewing only one part of some intersection and thereby have a poor perspective on the overall issue. For instance, atheists can face social stigmatization in the United States, but how social stigma affects people can vary widely by sex/gender, so examining the issue isn’t a bad idea as in https://gen.medium.com/women-nonbelievers-still-face-intolerance-despite-growing-numbers-672f95da8dde. But we shouldn’t pigeonhole someone as a-2nd-generation-Hungarian-20something-atheist-cishet-woman-from-Baltimore and think we get them.
And of course sometimes there are harmless ways to intersect--Asian Dodgers fans, or dog-owning Harvard students. But these things should be the exception, and be done in a positive context.
Otherwise, I think the concept of intersectionality is harmful: both dehumanizing and antagonism-provoking, compared to the idea of blending. Also, it's largely wrong (unless we decide it's true and act accordingly). It's kind of like firearm usage. Sometimes, often in dire circumstances, it's the right tool for the job, and sometimes increases safety. But a lot of times it just causes needless harm.
So, in summary:
(1) Addressing ideas over people is good, and ideas can be attached to reputation to help weed through the noise.
(2) Chopping things up into intersectional groups is (on balance) bad, and the alternative of blended backgrounds is good in comparison.