You're thinking of a treadmill used for running though. And I agree, for running that is the intent. But things are different here, the intent of the treadmill is to just match speed, but matching speed =/= negating speed in this case, that's the main hang up.
Again - the typical use of a treadmiil. Treadmills aren't typically used on planes as far as I know.
I don't disagree with the physics of what WILL happen in terms of thrust and the treadmill.
But I think the question is deceiving in that the intent of pretty much every type of treadmill is to keep whatever is on it - stationary.
I guess when I read that question - the image I have is that the plane is putting out all the thurst it needs to normally take off and the purpose of the treadmill is to keep it from moving forward - however that needs to happen.
And the myth is that somehow, someway that all the thrust magically provides some sort of lift for the plane to suddenly take off.
What this discussion is saying that you can't plausibly accomplish the feat of keeping a plane on the treadmill.