One of the problems theorists have in addressing and explaining large-scale inequity is that people tend to think in linear (or sometimes planar) terms. One-dimensional reasoning (e.g., the Left-Right spectrum) is natural, two-dimensional reasoning (e.g., Game Theory matrices) is common, three-dimensional reasoning is rare... But this tendency to break things down into one or two simple causal forces or factors usually creates a poor model of complex systems of interaction (like societies), and often becomes a focus for argumentative rhetoric.
Put simply, it's like the Three Body Problem in physics. The interaction between two people (like two orbiting bodies) can be quite predictable, but toss in a third and predictions go out the window. Toss a large, large number of people into a proverbial box (like thousands of pool balls rolling across a table) and patterns will emerge (the way patterns emerge from any non-linear dynamic system), but those patterns won't easily be connected to any single factor.
One has to reason about systems if one wants to understand society. But systems are tough to get a handle on.
The immediate problem with substituting socioeconomic status (SES) for race — i.e., SES profiling instead of racial profiling — is that any sharp thinker will turn around and ask: "So then why is it that blacks have such a comparatively low SES?". This instantly shifts the issue of profiling from police to the society as a whole, where society implicitly 'profiles' blacks into low-paying jobs and low-income communities, and then asks police to monitor those communities more stringently. SES profiling merely turns into second-order racial profiling. It's not an idea that can gain a lot of traction unless one engages in studied ignorance, refusing to accept second-order reasoning as meaningful. Which (unfortunately) seems to be a common practice in the US.
If you want to go the SES route, it's worth thinking about Immanuel Wallerstein's theory on the issue. Wallerstein placed the blame for what we now call profiling on the capitalist system, to wit:
- Capitalism as a system must enrich some at the expense of others; it's a system where some people win and some lose.
- This causes cognitive dissonance. It's difficult to look at an impoverished person and think: "That person is impoverished because I have wealth". It's even worse in the reverse: "That person has wealth because I am impoverished". The first leads to a pervasive sense of guilt, the second to a pervasive sense of anger.
- To ease that guilt/anger, a society will push an easily identifiable group into poverty. Then it associates poverty with that group, and all cognitive tensions subside because it's that group's fault they are poor, not our fault. They did it to themselves.
That way, instead of thinking "He's poor because I'm wealthy", we can think "He's poor because he's [X]", where X might be 'black', 'Mexican', 'Arab', etc. And the same works is reverse: instead of thinking "I'm poor because he's wealthy" we think "I'm poor because he's [Y]", where Y might be 'Jewish', 'white', 'Liberal', or any other group that can be identified as a nefarious oppressor. We lump groups into these social roles because it gives us an obvious external focus on which we can dump any sense of guilt or responsibility, making it their problem, not ours; we do this so that we can avoid criticizing (or even thinking about) the capitalist system that drives the whole process.
Racism then becomes a natural and mostly unavoidable expression of adventure capitalism; if we had no racial divisions before capitalism came to the fore, capitalism would have been forced to create racial divisions, else it would implode under a pervasive sense of unfairness. Remember that slavery itself was first and foremost and economic activity...
So, when try to wrestle with the question of whether police profile on the basis of racial characteristics (skin color, facial features, etc) or SES characteristics (clothing choice, apparent value of car, neighborhood, etc), the answer will almost always be 'Yes' to both. Our social system has arranged things so that the observable characteristics of race and SES correlate, and it has done that to protect the underlying system that causes inequity from any criticism or revision. Some cops are racist, some are classist, some are just bullies (and most are none of those), but that hardly matters. Cops don't serve justice, they serve the system, and the system depends on injustice for its survival.