That's exactly it. The plane is sitting on the tread, not the ground. As long as the tread is moving backwards at the same speed the jet is moving forward. The jet cannot move forward in relation to the ground. It is moving forward in relation to the tread mill.The ground is not moving, the surface of the treadmill is. As well as the surface of the tires. The ground only moves in relation to the body of the plane after the engines thrust it forward, or the plane puts on its brakes.
The car on the tread mill in the drawing above. Is turning it's wheels at the same speed as the tread. If you move the car forward on the tread. The wheels are spinning faster than the tread is. Even if it is only slightly more.
However the tread mill is designed to counter that movement. So it would speed up too. Your hand on the car is an outside force. If you continued to pull it forward the tread mill could accelerate to infinity and it couldn't pull the car back. You are moving it forward.
If the car is moving under it's own power. It has to mantain the speed it is moving on the tread, on it's own. So if the engine tries to pull the car/plane forward. The tread mill speeds up to compensate. Again resulting in zero net movement.