"Driving a horse to the bit" actually shortens the frame. That is the idea we are getting at. On the bit is a starting point, and eventually in collection - the horse shortens it's frame immensely.
Think of the horse's body as a slinky toy. One end is the head, the other is the hind legs. When you put leg on, you are driving one end towards the other (the head). By applying gentle restraining aids (bit, seat), you can contain the energy so the slinky condenses, and doesn't flip down the stairs loosing all of it's energy ;)
The shorter/more condensed the slinky is, the more energy it has to use.
That is how we get a Dressage horse to perform movements such as the Piaffe. We have condensed the 'slinky' so effectively, that the energy created can be used to created such a spectacularly collected movement. It is impossible to get such a movement out of a horse that is strong out - they just do not have the pent up energy to hold it.
Reading your posts, I don't think you should even be thinking about putting your horse 'on the bit' at this point.
She needs better basic education, and you will be surprised that her way of going will improve simply with better reactions to the aids.
If she shoots off every time you put your leg on, get her used to leg. Yes you want a sharp reaction to the leg, but not running away from it. Squeeze your calf at halt to ask for walk. If she shoots off, bring her back to halt, and repeat. Rinse, lather, repeat, until she realises that she just has to step off quietly and the pressure will come off. Same process for other transitions.
Get the basics right before you start fussing about anything else.