Youre right, we are all hugely multifaceted individuals. It is hard and maybe even weird to try to isolate the notion of privilege down to a single domain, but often, this is what science is trying to do: control all but one variable and try to observe and measure an effect.
¯\_(ツ)_/¯
We can, for example, try to measure the effect of one carcinogen to cancer risk. We can’t say all cancer is attributable to one carcinogen.
To relate it to the topic at hand, you would have to measure to total variation of privilege attributable to race as compared to other factors like nationality, socioeconomic status, hell even height, weight, and physical attractiveness. In other words, it’s an incredibly complex model that shouldn’t be reduced to a single variable to estimate effectively.
So, to extend your example, we can't necessarily say p(successful outcome|white)*p(successful outcome|poor) > p(successful outcome|nonwhite)*p(successful outcome|wealthy) until we measure those other socioeconomic conditions. It gets more complicated when we consider they may not all be independent of one another.
And that's just one extra variable. As you say, most agree we are multifaceted individuals. Adding dimensions like p(successful outcome|disability) or p(successful outcome|female) further complicates this as do all the other variables of a social species. So we can't really make strong statements about outcomes of a system very well unless we measure across the entire system of variables. I think that's what people are poking at. We may be able to make conclusions about individual variables but that doesn't always translate to an accurate model of the system at large. It's interesting and perhaps useful, but certainly incomplete. It only works under a "ceteris paribus" assumption, which is to say: "it only works in contrived circumstances that don't reflect reality."