...I used this for a long time.
Much modified Raleigh Sports.
But lately, I find myself using something like this more. It allows for a slightly quicker ride, so I can take it out along the bike trail, do some distance, and still stop on the way home to collect some few items we've run low on, buy a fresh loaf of bread, etc.
The only drawback is if you are worried about theft while you're inside the store, as problematic in your area. I carry a couple of U-locks, which kind of slows me down on the longer ride. But your initial surmise is correct, almost any bicycle can be adapted toward errands and grocery shopping.