"If swing is moving from a fixed point and the fixed point also moves. Then there is a time when the fixed point stops moving therefore we introduce sway."
First, waltz swing is moving from a fixed point which DOES NOT MOVE during a given swing. You dance one complete downswing-upswing ccyel about a fixed point, then get a new fixed point for the next swing. Think about tarzan - swings on a vine from one tree, then swings on the vine from the next, but each vine remains fixed to the same tree it started on.
Then, swing and sway are not perfectly coordinated. Consider that downswings do not have any say. Swing is the path of the body, while sway is an incline of the body - the body can move in a swing path, without have any sway.
However, it is indeed generally beneficial to use some sway on the upswings. For one thing, sway lets you extent your free foot further down the floor, without it getting ahead of your body, because the incline of the body means that a point "under" the body (along its axis) will be further down the floor than "under" the body with respect to gravity is.