"Witches fly on broomsticks."

FALSE.

Witches fly on airplanes.
The idea of flying to meetings on a broom is derived from the use of 'flying ointments'. These often poisonous ointments used to be made to heighten the senses, much in the way that some people today use drugs. The effect of some of the ingredients was to give the illusion of flight.
Most Witches discourage the use of artificial aids to increase awareness, partly because such use can be dangerous and partly because if awareness or heightened senses are acquired or enhanced with drugs, they are not truly your own.
Witches do, however, use the broomstick to sweep, whether practically as in removing leaves from the porch or garden or metaphorically in preparing a working space.