I'm not really getting why it's so important to ditch the word "intersectionality." Similarly, there are many folks who have a problem with the term "feminism" and would prefer the word be dropped in favour of just following its concepts. I personally don't see why losing either word is necessary. For me, intersectionality describes the concept in a way that "multiple oppressions" or "affected by sexism and other forms of oppression does not."
If you read Crenshaw's original concept of intersectionality, it isn't simply about one person being affected by more than one form of oppression. It's that the experience of oppression in in itself is different where two forms of oppression intersect. One example in her 1991 article (from memory) was that of workplace discrimination against women of colour in the US. Say for example, you were on the receiving end of comments relating to the stereotype of the highly sexualised, "exotic" woman of colour. The perpetrators could be either men, women or both.
You would have no redress under race equality legislation because that wouldn't apply if other employees of colour weren't similarly affected (and they wouldn't be if they were male.) You would have no recourse to sex discrimination legislation because other women in the workplace were not similarly affected (if they were white.) Your experience of harassment and discrimination is in the context of being both a woman and of colour.
The concern expressed by many women of colour is that they are regarded by feminists as women, who also happen to be affected by racism. Similarly, they are regarded in civil rights/anti-racist groups as people of colour, who also happen to be affected by sexism. Neither of these reflects who they really are and quite rightly, they may be wary of the capability or inclination of either type of group to represent their interests. They are also rightly wary of experiencing discrimination within those groups.
It's possible to translate this example to other people who are affected by intersecting forms of oppression.
So, just like throwing out the word feminism runs the risk of losing sight of what the concept is about, throwing out the word intersectionality runs the risk of losing sight of that concept as well.
I'm also a bit wary of the focus on "imutable" characteristics vs "changeable" ones with regard to people affected by oppression. It still suggests that the former somehow sit higher on the ladder or carry more weight than the former, and I can't buy that. For one, it's rather insensitive and patronising for someone who doesn't experience a characteristic that results in oppression (e.g. disability, Lesbian, gay, bisexual, trans, etc.) to pronounce that their characteristic "can't be so bad" because it's something they can change, or others can change perceptions of.)
Also, if there is an argument that there should be greater "solidarity" between women because they share a sex, surely the same would apply to people of colour, because they can't change their ethnicity either.
And, where would that then leave women of colour? Torn in half? Forced to choose between being a woman or being a person of colour? I understand that is what many already feel.