And how does the kite get to this altitude where the winds can't suddenly stop?
In my experience flying small planes and gliders up to 10,000 feet, I've never flown at an altitude that the winds can't suddenly stop or change direction. It was especially important on long flights in the glider over flat landscapes that this was not the case. And I've never heard of anything lower than the jetstream where light winds were impossible, and I don't think these kites will get to 30,000 feet in their 45 second flights.
At altitude, air is less dense and can move faster. But this is not the same as the wind at altitude moves fast. There are less objects to cause instabilities and changes in direction, but this does not mean it can't. Wind shear exists at every altitude planes typically fly. Even on days with a steady prevailing wind, for every thermal moving air upwards, there is a cylinder (-ish, shape may vary) around it of air moving downwards. These changes of direction matter, too.