Want an example of why the treadmill acting on the wheels will have NO effect?

Go into an airport...you know those moving walkways?

Pull the suitcase with wheels on the normal floor. Now, pull that same suitcase on the moving walkway (against the movement of the walkway).

Do you have to exert any extra force? Nope. You pull with the same force. The wheels of the suitcase on the treadmill, however, just spin faster.