Food prices have really shot up since 2006 though; in particular, bulk beans have almost doubled. Still, I think it would be possible to eat the same diet today for around $35/week for a couple.
I can easily imagine eating for around 2 Euro a day or even less. It just would not be as much fun. So I get around with 5 to 7 Euro, that allows me to add delicious ham and some other meat to my diet in small portions.
My point is that no one can honestly say that they are fat because they can't afford good food. There are tons of excuses, but it comes down to personal responsibility.
I heard a story about a McDonald's closing in a poor area. There are at least two effects of this, the second of which was quite surprising: 1. McD, a source of high-fat food, leaves the area, and subsequently people need to find alternative food; 2. McD, a source of cheap food, leaves the area, and subsequently people have trouble affording food.
Apparently McD's distribution chain is so efficient, that their presence resulted in wider access to food. I say nothing about nutrition, jobs, or anything else. Just food that fills your stomach. It seems like nothing is that simple enough to summarize in a single sentence. Doing so is likely overestimating of your grasp of the situation. But perhaps it is indeed just laziness.
(I say this as I'm about to re-heat some left over spanish rice that I made last night. It cost almost nothing to make.)
So it's not exactly that they "still earn enough money" to become obese, it's that somehow the food supply has been inverted so everyone can afford calories but not so many can afford healthy food.
Besides that, as a counter argument to your hypothesis, wouldn't inner city neighborhoods be old and walkable? In that case, your argument that they are more healthy because they are inhabited by richer people doesn't seem to fly.