Grocery Stores closing really is happening, and really is a problem. And it's part of what's killing towns - if only because attracting new folk without one is hard.
We live in John Day, Or (pop 1700)(County pop 7200, area: about 2x Delaware), which we chose partially because of it's supermarket. The next closest supermarket is 90min away. (Walmart: 2h). Not having to commute hours for groceries is a big deal, as is being able to get vegetables of season - and in the rural west these things not a given.
When looking to relocate from Seattle to rural town in the west with hospital (wife's a doc) and broadband (for me), it soon became obvious that the quality of the food available was going to be the next most important thing on the list (and much harder than broadband). Too many places we looked had long commutes to groceries - or stores with almost no fresh food‡. Most places have no farmers markets or farm stands outside of summer, and early fall. And for us at least - living off of canned and frozen food just isn't fun (at least if we've not canned/frozen it.)
This really restricts immigration which is critical economically when your town's youth mainly move to the city as is really common. These small towns are dying a death of a thousand cuts, but loosing your grocery store is a pretty big cut.
‡ My favorite story here is once when living in a small town with 2! stores for a short while, we decided to have a dish that called for cooking some spinach. Neither had fresh spinach, nor any substitute - kale say. Neither had any frozen spinach, the second store did have canned "creamed spinach" - and an ingredients list that included both condensed milk and corn syrup. We made something else.