You got it backwards, witches didn’t ride broomsticks, the witches fly under their natural power and carried the broomsticks with them.
Broomsticks are the most persecuted and hated of all religious practicioners. From earliest days, hate-filled and prejudiced housewifes (and others) grabbed broomsticks, suddenly, around the throat, and rubbed their arses on the ground furiously. Thousands of broomsticks were beaten to death in this way.
Treated as slaves and worse than slaves, broomsticks were relegated to the darkest closets, were not fed with the family, were kept naked and subservient. Disney’s FANTASIA has a scene where Mickey chops a broomstick to pieces, and audiences even today laugh and cheer!
In the Middle Ages, a few good-hearted wiccans tried to save broomsticks from this involuntary slavery, by rescuing them and flying them to safety. Wiccans can fly under their own natural powers, although they usually keep this secret. The nasty, broomstick-hating Christians saw the witches flying away with the broomstick, and attributed the power of flight to the broomstick! This only engendered more fear and distrust, leading to the great broomstick fires of 1542 in Milano, Italy, when thousand of broomsticks were burned at the stake. The sight of all those broomsticks, tied to the stake, with straw all around them to ignite the fire… would have brought a tear to your eye.
In some locations nowadays, broomsticks are actually extinct or have been replaced by lifeless plastic copies.