I would imagine that it's not just convenience stores. I frequently ponder the fact that the grocery store closest to me is a marque of a US conglomerate (I live in Canada) that manages to have "low prices" by just treating Canadian stores as part of its US logistics chain—almost all of its products are shipped here from the US, rather than locally produced under US brands. And that includes things like baked goods and produce.

Thus, everything perishable in that store is either much "older" than you'd expect (i.e. it will go bad very soon after you buy it), or is pounded with preservatives—the bread, bagels, muffins, etc. all sort of have this uniquely-bad "springy yet doughy" texture: the sort of texture that I associate with Wonderbread.

Within the same distance there are local bakeries, produce markets, butcher shops, etc. that sell good stuff. But I don't doubt for a moment that certain families ignore those and go straight for the "everything shipped from far away" supermarket for all their shopping—and are having different health outcomes because of that.