The "pen and paper" analogy is an excellent description which would obviously work in the vertical as well, except it would not be possible to demonstrate.
As for the inertia part. The aircraft's mass/weight is defined/affected by gravity, not by the surrounding air parcel.
Therefore if the "pen" is replaced by half a matchstick, representing a very light aircraft rather than a heavy jet, when the "gust" is applied the very lightweight aircraft will react quicker (catch up quicker) it's airspeed will change relative to the air parcel, but by a smaller amount and for a shorter time than the heavy aircraft as the inertia, body at rest etc, takes over.
An excellent discussion and difficult to explain.
Perhaps if the parcel were to be a bubble of air suspended in a vacuum, then remove the bubble forwards and the aircraft has no air to keep it up and will fall through the vacuum until another bubble comes along in the right direction to replace the lift. Ground speed has nothing to do with it, until the bubble is replaced by a bubble of granite which produces very little lift!
Please excuse my gibbering.