Places are food deserts primarily because the inhabitants choose not to pay for healthy food. Providing access to food does not change people's consumption choices:

http://theincidentaleconomist.com/wordpress/giving-people-ac...

(There are lots of natural experiments too - e.g., Jersey City, where I used to live, had lots of fat poor people and lots of svelte yuppies. Strangely, the poor didn't start eating Kale when the yuppies moved in.)

Assigning the label "victim blaming" to an idea does not prove it wrong.