Alright. Picture this, Chumpman.

You've seen a car on a dyno machine before. So, no matter how fast the car goes, the rollers of the dyno will keep the car in place, because they spin at the same speed as the tires.

Now, imagine a jet plane on a dyno. You put all of it's wheels on the rollers, fire the engines and the rollers would not spin, the plane will propel itself forward going over the rollers because the power does not come through the wheels. Thus, the speed of the treadmill is irrelevant.

What's going to end up happening, is that the wheels of the plane taking off will spin at twice the speed they normally would, given any speed the plane reaches during take off (since the treadmill mathes the speed of the plane, but in the other direction).

Plane lifts off with the wheels spinning twice the speed they normally would, that's all.

A treadmill moving in the opposite direction of the plane, does provide enough force to counteract the thurst of the plane.

If it did apply enough force going in the opposite direction of the plane to stop it from lifting off, then it's logical to think that if it treadmill was standing still, it would provide HALF of that force? According to your logic it is so, however it's not. Think about it.