There is a very simple answer to this problem: VEX.
Over the summer, Team 116 built a VEX demonstration field (a very simple 8'x8' table covered with neoprene floor pads) and a set of VEX "squarebot" robots. We created a very simple 2-minute game that takes 15 seconds to describe, and then turn over the controls of the VEX robots to anyone that wants to drive. We have had drivers as young as four, and as old as ... well, lets just say they were a lot older than anyone on the team. We have demonstrated it at many public events (the Herndon Festival with over 90,000 visitors, the NBC DigitalEdge Expo with 70,000 visitors, several area school demonstrations, etc.), and well over a thousand different people driving the robots. The relatively low power and mass of the VEX robots (vs. an FRC robot) limits the amount of actual destruction they can do, and we have had absolutely no problems with the public damaging the robots. To the contrary, what we usually have is a reaction like these:
We have the team FRC robot to one side - unpowered - where the team members can explain the program, describe the robot, and not let it hurt anyone that does not understand it.
As an aside, I would also point out that our robotics team captain is a cheerleader.
-dave