The original throught experiment is:
"A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?"
As pilots we know that flying works on airspeed and not groundspeed and that airspeed at takeoff is from thrust being greater than drag so ... why worry about a runway thats moving?
The misapprehension that confuses 'people' - not pilots, we are not mere 'people' - is that you link in your mind the conveyor belt moving against you to when you walk the wrong way on a travelator (or against the travel of an escalator) and can 'stand still' with reference to the world around you while walking.
The probelm applying the walking view of standing still to an aircraft taking off is that unlike feet being carried backwards at the same rate as the forward steps the tires of the aircraft are able to spin provided there is thrust equal to the drag of the aircraft on the belt ... the belt runway does not carry the aircraft backwards at all if the engine is delivering take-off power ... it just increases the drag of the spinning wheel bearings ... and that drag is VERY small compared to thrust from engine/prop - thrust > drag = accelleration.
So the pilots answer is ABSOLUTELY YES the aircraft will take off. It matters not if the belt runway is trying to 'pull' the aircraft backwards at takeoff airspeed or even greater speed ... all that happens is the wheel in contact with the belt runway will spin up to a speed equal to takeoff speed + belt speed ... and so long as that spinning does not disintergrate the wheel you are fine.
If you turn the throught experiment around and make the converyor belt run in the same direction as the aircraft your wheels will spin at airspeed - conveyor speed = 0 but you will still reach takeoff airspeed and it will be fine ... and youtube provides many examples of this when aircraft take off or land on moving trucks ... they fly on/off at airspeed but the aircraft tyres do not rotate.