The second point I want to make is that you have a mis-placed emphasis on "fresh food". Frozen or canned vegetables, fruits and legumes are actually more nutritious than fresh produce. And the cost, variety and quality of canned and frozen produce has been improving steadily for decades.
The food desert hypothesis sounds compelling, but simply doesn't jibe with the evidence. Obesity among the lower classes is almost certainly driven by cultural and/or genetic factors, not lack of access to healthy food.
[1] https://www.npr.org/sections/thesalt/2015/03/20/393943031/wh... [2] https://www.vox.com/2015/12/22/10644738/food-desert-evidence