What I'm thinking right now is a bike on a downhill ride. The propulsion is due to gravity, you're not pedaling, but the bike's wheels still spin. Technically, the wheels are freely spinning, but the bike is moving because the wheels are moving. The wheels can be spinning "perfectly" in which case the point of contact between the wheel and the ground has a speed of zero, or the wheels can be sliding. In either case there's friction: in the first case, it's static friction, and in the second case, it's kinetic friction. Both will tend to retard the motion of the bike.
A plane in contact with the ground is the same way. It doesn't matter where the propulsion comes from: whether it's the propeller, jets, or a giant truck that's pulling on the plane to the hangar, or a dozen crazy guys with sturdy ropes pulling on it. The plane moves because the wheels roll on the ground. Once the wheels start rolling, or even sliding, there's friction, which means you have to pull harder to get the thing going.
The only time that the conveyor belt will have no effect on the motion of the plane is when the plane is not in contact with the belt, in which case it's already flying anyway, or when the belt is frictionless. My assumption, in my long posts, is that the plane stays in contact with the belt and the belt is not frictionless.
I'll concede that once you assume otherwise, then the plane will take off. But at least I can now see how and why.
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