Well, flying would mean that she'd have to sit on the broom. They could have harnessed her up, but then she would've been flying -and- concentrating on making the broom look good -and- steadying herself -and- belting. OR they could have suspended the broom, and...I wouldn't trust the broom, especially because it's hollow and extremely light. I just think it would be too hard.
I love her flying machine, if we shall call it that. The way she waves her arms looks like she's soaring in the air. I don't think it's cheesy at all, and I would rather have her do this than fly all over the place. At least this way, we get a good look at her singing.