So now the treadmill is unpowered free to move and you push the vehicle body to the right. Of course since you apply a force to the right to the vehicle body the vehicle body will move in that direction.
OK, well that's a start!
But it has basically nothing to do with the original problem where the only applied force is to the left (not to the right) and it is applied at the input wheel.
Do you want to learn that it does indeed have something to do with the original problem, or must you hide under your safety blanket?
Now, imagine the right wheel is not connected with a belt (and, sorry, I used 'horizontal belt' previously when I should has used 'treadmill' to save confusion). The wheel is locked rigid. You, the HoG or whatever push the vehicle to the right and that causes the treadmill to move to the right. I presume you are perfectly OK with that since it's pretty basic. So..
Let's say the vehicle moves to the right (via whatever strange force) at 2m/s. With a locked right wheel the treadmill will move to the right at 2m/s. Yes? Good. Clearly, any clockwise rotation of the right wheel will reduce the speed of the treadmill. We can thus have three distinct situations:
1. The connection between the wheels is 1:1 (that is, the left wheel drives the right wheel at the same speed). In this case, the vehicle moves right at 2m/s, the right wheel rotates at the same speed the vehicle is travelling and the treadmill is stationary.
2. The connection is 2:1 (that is, the left wheel drives the right wheel at half the speed). In this case the vehicle moves right at 2m/s and the treadmill moves right at 1m/s (since the right wheel is rotating slower than is necessary to keep up with the vehicle speed).
3. The connection is 1:2 (that is, the left wheel drives the right wheel at twice the speed). In this case the vehicle moves right at 2m/s and the treadmill moves left a 1m/s (since the right wheel is rotating faster than is necessary to keep pace with the vehicle speed).
Do you agree with those? If not, which of 1..3 do you think is wrong? I can assure you that this will resolve your original problem if you stick with it.