Nitpicking aside, I suspect that everyone agrees that in order for the plane to take off it must move
forward along the conveyor.

Which is exactly my point! If you have a motoroized conveyor which
always reduces the plane's forward movement to zero, no lift is
generated, preventing the plane from flying. In other words, the plane
generates lift by moving air over its wings. It moves air over its
wings by moving forward. If you zero out forward movement, by a
motorized conveyor, resulting in a zero delta, no lift is generated.
it's a question of the plane making forward movement.

Specifically, as it relates to your reply, while prop wash would indeed
produce some lift over the wing, it would not be nearly enough to
obtain take off....which is why we have wings. Thusly, if forward
movement is zero and you're full throttle, you're not airbound....which
is exaclty the same thing as a plane with no wings.