The word coach comes from the Hungarian village of Kocs, pronounced “kotch”, made famous for its large, horse-drawn carriages in the sixteenth century.
In Britain, the word became coach, and by the nineteenth century took on the second meaning of a sports trainer or private tutor.
The implication is that, through his experience and knowledge, the coach, like a bus or a train car, carries the younger trainees to their destinations.
Both the manager of an athletic team and a large passenger vehicle called a “coach”.