While I understand that there is a relationship between poverty and poor health, I see the relationship as a correlation rather than a causation. And as you have noted, more than money comes into play to make the difference.
OK, then, what do you see as the causal relationship? If X is correlated with Y, then either X causes Y, or Y causes X, or Z causes both X and Y. Or some mixture of those.
Does poverty "cause" obesity? Not directly. A rich person can eat poorly, avoid exercise, etc. A poor person can eat healthily, exercise, etc. As I've said.
However, poor people, as I've also said, 1) have a harder time accessing healthy food 2) are less likely to be well-educated about healthy eating 3) may have a harder time moving around their neighborhoods 4) live in a culture in which heavier weight is more socially accepted ... and so on. There are many factors. My point is that poverty does cause those things I just mentioned, and those things cause obesity. There's just little room for argument there.
Since we're sharing our own experiences, let's take my middle-class situation. I have two upscale grocery stores within five minutes' drive of my house that, needless to say, have excellent produce and meat sections. I have a lower-priced grocery store 10 minutes' drive away that has pretty decent produce and OK meat. If I take my dog on a walk, I can walk 15 minutes in one direction and come to a path around a quiet lake, or 10 minutes in the other direction and come to a walking/biking path along a wooded creek that eventually leads to a big park with a waterfall, or 15 minutes in still another way and come to three lakes that are linked together by bustling biking and walking paths and parks, including a rose garden, a Japanese peace garden and a bird sanctuary. I can walk around one lake in about an hour, or bike around all three in about that same time. In between these destinations are houses -- some of them, like mine, smallish and others literal mansions -- many (of both sizes) with beautiful landscaped lawns and lovely gardens. In 10 years of walking around my neighborhood, I've never felt unsafe except once or twice walking alone in the dark and encountering someone -- but even then, nothing happened, all was fine. For years, I subscribed to
Cooking Light, and even now tend to look through healthy recipes in other magazines or the newspaper or the web. There's an Anytime Fitness five minutes in one direction, a Snap Fitness in another, a YMCA 15 minutes away, and so on. And yes, there is some cultural pressure to stay thin -- more, I would guess, than in some neighborhoods but far less than, say, in Hollywood.
So how many of those factors that facilitate and encourage exercise and healthy eating does your hypothetical resident of da 'hood enjoy?