"The object's velocity in its circular path is constant. There is no acceleration in that respect."
Isn`t acceleration defined as the rate of change of velocity. Velocity has direction as well as magnitude so a change of direction is a change of velocity which has an associated acceleration.
Your experiment with the rock shows that centifugal force is not the reason the moon doesn`t crash into the earth. If the force you describe in the earth moon example worked the way you claim it, would need to act in exactly the opposite direction to the gravitational force. Since the gravitational force is equivalent to the rope in your example that fact the rock did not fly away from you in the direction the rope was pointing in surely invalides the idea of a real centrifugal force.I bet the rock flew of at a tangent to the circle it was being whipped around in. The moon is in freefall around the earth as described in your post. It is constantly accelerating but it is the direction not nescecarily the magnitude of the velocity that is changing. Unless the object speeds up or slows down or moves through some lines of equipotential there isn`t any transfer of energy. The accelerating car is different because you are speeding the vehicle up (increasing the magnitude of the velocity) thus adding energy.