If they specialize they can find a niche that the large players don't cover. I have seen a few markets that were not specialists. They had the same American food selection that everyone buys (obviously in different areas the exact selection is different, but within the region none). The same name brand spaghetti sauce (but only of the 2 name brands, not the 10 in a supermarket), onions, ground beef you buy in the large supermarkets today, but at higher costs.

The Kroger chain you visit killed all of the above (except for the gas station convenience stores which are a smaller version of the above)