I believe fundamentally people are ill equipped to deal with this kind of complexity. When labeling people, rather than using a small set of discrete terms, it's closer to a probabilistic clustering problem where each individual is represented by a high dimensional vector where the clusters are generated on the fly and per-use, parameterized by time and space and the observer's viewpoint.
This complex representation is what the area of academia that studies identity has come to call intersectionality. I.e. being black and gay is a particular experience that is also modified by the person's other demographics.
But imagine translating that level of complexity to national discourse (i.e. a CNN article). Doing so leads to two simplifying approaches: one is to attempt to simplify by using broad categorizing terms ('black', 'white', etc); the other is to attempt to reject those terms by not using them. You're describing problems with the first approach. The problems with the second approach are that it can sweep actual experiences under the rug. For example, while 'black' is an incredibly broad term, that is in fact how a large portion of the population labels another large portion of the population, irrespective of their background, which does create a shared black experience that can and should be talked about, but then always at the risk of ignoring other labels that also make sense in context, such as wealth, education, etc.
I don't know a rhetorical way out of this situation. This is why academic texts (by that I mean authors who are attempting to tackle the subject without any undue attempts to simplify) dealing with the subject can become so twisted and hard to read, because actually describing the context of an individual person can lead to a per-person book-level explication of their experiences that also includes a not-insubstantial explication of the author's experiences.
This is a problem with language. Every time I'm personally in a situation that ends up getting written about in the media, a situation where I'm sufficiently involved to understand the nuances of what happened, the article feels like a comic book simplification and reflection of what happened. But then did I experience that event in the same way that other participants did? No. That's the subject matter tackled, for example, in Roshomon. An 'event' in history is a generalization and narrativization of something that will be probably be interpreted very differently by the actual participants involved.
The problem with exercises like what I just wrote is to use the lack of conclusive and easy categorization to justify inaction or disengagement, i.e. "There is no black identity, let's just treat everyone as though they're the same race." That flies in the face of a great deal of lived experience. Especially because a white person in America can afford to act that way, but a black person cannot.