I think "it depends"...
Are old (uninsulated) basements in your area damp and musty and gross? Or are they, more often, dry
enough? If they're not all gross, I'd say you don't need insulation in that corner.
Where in the world might you be? One reason to not insulate is to harness the coldness of the earth below the basement. But if your earth temp 6' deep is warmer than a
root cellar, insulating might help you keep it colder. Even in WI where my basement root cellar is, the soil temp below my basement floor is likely 45 degrees in winter, warmer than the root cellar itself. My cellar gets cold because part of the basement wall is exposed above grade and not insulated so it gets very cold. That, plus some forced ventilation on cold nights, helps my cellar get cold enough in October to start storing things.
If the floor doesn't get insulated, one concern may be the high humidity of the cellar rotting out the nearby house parts (floor joists, etc). So you'd want to build the cellar so that humidity from it can't get out into the basement. Vapor barrier would be one way...
If you do insulate it, you should be fine as long as you can get cold air in there in the autumn. Regardless of the insulation detail, I'd put a fan in one of your vent pipes so you can force cold air in when the temps outside are favorable.