Originally, especially in Europe, a broom was exactly that! Broom!
Broom is a springy bush that grows in "copses". For hundreds of years (Possibly thousands!) farmers have used Broom to make fences.
A "sweeping" broom was used only to clean the house! It is only in later years that the broom has gained some "magical" power, apart from its early use in marriage. Before Christianity took over,the young couple would make their vows (with meat and wine), and then hold hands (hand fasting) to "jump over the broomstick", to signify that they were about to "keep house" together.
Here in Yorkshire, when a couple are living together but not married, they are still referred to as "Living over the brush".
That was the originally name for Broom; brushwood!