While I don't have a family myself, this sounds great! A lot of work, perhaps more than you realize, but what a way to create a supportive atmosphere for younger folks to learn skills! Hopefully this would instill a love of mountain biking for life, or at least physical activity in general.
You could have a contest of some kind to choose the name of the group Would you set up special "girls only" and "boys only" rides to go along with the scheduled free rides?
Looking forward to learning what you decide to do and how it progresses!