The air surrounding the merry-go-round is stationary where as the earth's atmosphere rotates with the earth. If the air surrounding the merry-go-round was rotating around the same point, would the ball still veer?
The ball would still veer because it has inertia, but it would not veer to quite the same extent because the rotating air would try to accelerate it in the opposite direction.
However, you are slightly missing the point of the video. The ball being thrown, IS the air moving on the Earth. You are right that the atmosphere is part of the Earth and is rotating with it, but WIND and movement of tropical storms, is movement of air RELATIVE to the Earth's surface - just like the ball in the video.