The skier actually has a great deal of control over the path taken, because they (indirectly) select the angle of the rope at each moment in time.
However, both some analysis I've done and some overhead shots suggest that the skier does indeed *prefer* a path that is longer as the rope gets shorter. The easiest way to see this is that the skier travels the same distance across, but increasing distance in the downcourse direction. They are basically taking a less direct route to the ball.
And at very short line lengths, deviation from the easiest possible path gets much harder really fast. So probably nobody will be choosing their own crazy path -- it's just too difficult.