When you live in a densely populated area, it is a near certainty that you will have a supermarket well within walking distance; no more than say, 10 minutes away by foot. At those distances you don't really "shop for your weekly groceries" but rather really shop for a handful of things at a time, whenever you need them.
You are putting the dogcart before the dog. For a variety of reasons, land use patterns in Europe encouraged keeping as much rural land as possible as rural rather than urban or suburban. I would posit that there wasn't any greater desire to live in densely populated areas in Europe than America, but rather that those in rural areas had both the desire and ability to keep them rural. NIMBYism is responsible for land use patterns on both sides of the pond, it just is than on one side it has served more to keep suburban areas from becoming urban while on the other it helped keep rural areas from becoming suburbs and exurbs.
