Isn't causality transitive though? It sounds like you're saying that low socioeconomic status causes poorer access to healthcare and healthy foods, and that those cause worse health outcomes. Yet you're claiming that low socioeconomic status doesn't cause worse health outcomes. That seems wrong to me.
If causality were transitive the phrase “correlation doesn’t equal causation” wouldn’t exist
Surely that's incorrect. The most obvious scenario is A causes B, B correlates with A, but B does not cause A. Whether causality is transitive is irrelevant.
The quote is typically brought up when there isn’t a direct causal relationship between two variables, not when the causality is reversed. e.g. ice cream sales and drownings. In both cases heat drives behavior, but neither cause each other.
I’d say reverse causality is a very common example, particularly in health and medicine (e.g. illegal drug use and psychiatric disorders).