Pushing off refers to forwards, longitudinal force. Walking and running are commonly modeled by inverted pendulums, and the best time to increase the energy of the pendulum is at the extreme of the range of motion. The middle of the swing is the worst time to add energy to the system.

Or think of it physiologically. If you are adding energy mid-stride, the gluteus maximus is responsible for producing all forwards force. (Which might be a fair assessment, humans's are unusually large for mammals.) Adding energy at the end of the stride allow the gastrocnemius to provide thrust as well.