"It is a lot more expensive and needs special care and maintenance, yes. Wood is the traditional material for a good reason. It's cheap and durable if properly made. Steel is much, much more conductive to magical energy, which changes a lot of things about how the broom works. But for now I need to explain the basic theories behind broomcraft. To your seats, everyone!"
Why witches can do the things they can do is not really understood, but what is happening in a broom is mana - magical energy - being converted directly into force through an object. Witches can channel mana into some objects and it will just sit there, but to fly they push mana through a broom. They can do that for object they touch if they get a feel for it! If you were very, very good you could fly a chair!
However, 'flying' anything too un-broom-like usually just results in the thing flying off in some random direction, being smashed to splinters, or being crushed as it is pushed in every direction at once. Something about the 'broom' design makes the force easy to control, to channel down, back, left and right.
Understanding pouring mana through objects as innately as possible is a key skill for broomcrafters. There are math problems and diagrams and theories they will learn later, but the first step is to practice doing it and get a good feel for it.