As the kite moves some kinetic energy occurs, whether going up or down. Some similarity can work with Solar balloon jumping - #2 by Windy_Skies when the balloon rises before lifting a person by its kinetic energy.
The potential energy has to fully do with kinetic energy only when the gravity fully occurs.
Aerodynamic forces are higher compared to gravity. And there is a significant difference between the 1 ton M600 and 20 or 30 kg TU Delft wings, leading to a huge difference of kinetic energy, taking also account of M600 speed being the double (so the energy is squared) _ by aerodynamic force, not by gravity _ of that of TD wing.
In the paper in reference there are some measures with gravity incl. and excl. and they are right.