oh. my. god. Ok for the last time, there is no correlation between you "running" on a treadmill and a skateboard "rolling" on the treadmill attached to a rope. Just like there is no correlation between a plane "rolling" on the conveyor and a car "driving" on the conveyor. The car's forward momentum is derived from a force exertion of the wheels on the surface: the plane's forward momentum is derived from the thrust from the engines.
To make your analogy more accurate, and for your sake I would recommend actually conducting this experiment:
Go put on roller skates (or roller blades, whatever)
Stand on a treadmill
Tie a rope to the handle bars
Grab the rope
Now, speed the treadmill up really fast. Obviously you aren't going anywhere cause you are holding on to the rope, right? Now, pull yourself forward. You move forward. AND YOU WILL MOVE FORWARD NO MATTER WHAT THE SPEED OF THE TREADMILL IS. This is even more true of the treadmill is only going the speed you are pulling yourself. If you can pull yourself forward when the treadmill is going really fast, then you can pull yourself forward when it is going much slower, say, at the speed you are pulling yourself.
If you don't see this then you are just plain ignorant.
Now, apply this to the plane. You pulling on the rope is an analogy to the thrust of the plane. Your wheels on the treadmill are an analogy to the wheels of the plane on the conveyor (almost the exact same case). When you pull, you go forward, no matter what the speed of the treadmill is doing. Thus, when the plane turns on it's engines, it is "thrusting" forward (think of it as a plane pulling on a big rope in front of it) and it moves forward, no matter what the speed of the conveyor. Forward motion relative to the air will eventually create lift
The plane takes off.