If you do that, and ignore both wind and air friction, then the balloon will keep moving right along after you stop the boat. (assuming it is weighted so it goes neither up nor down after you cut the string, of course.)
I don't see how these moving vehicle analogies are helpful. The atmosphere moves with the earth because there is nothing to make it not move with the earth. It is that simple.
Let's stop using analogies with vehicles that move in the atmo. The confusion around whether or not air resistance factors in seems to be continually obscuring the thought experiment. If the atmosphere were not on average moving as fast as the Earth, then we would observe an average Easterly flow of air globally (from the East toward the West). We don't. Why again do you think the atmo would not move in concert with the ground?
Well I do see the clouds moving from California to Illinois , If the clouds would not be moving Illinois would move would more under the clouds produced at the shores of California I know the clouds belong to the earth is part of the system but they are two separated masses separated by a lower density masse which is the atmospheric gas not anchoring the ground and the clouds.
Right. Those clouds are actually moving prograde i.e. in the forward direction of Earth's rotation. The Earth, as seen from above the North Pole, rotates counterclockwise (New York rotates into dawn before California rotates into dawn), so those clouds are actually moving faster than the spinning Earth.