Okay, dumb question from someone who rides a bike but is not a "cyclist."

Road bikes have a pretty high tire pressure, to the point that the tire itself basically offers no cushion at all, on it's own. Why are these tires pneumatic at all? Why not wrap a rubber donut around the rim and call it a day? I'm sure part of the answer is weight, but surely this can be reduced either by making the rubber relatively thin, or using some sort of hard closed-cell foam underneath the rubber layer?