I hear ya on this. My daughter will be 10 soon and rides off and on just for fun with lessons and/or trail rides. She wants to start riding more frequently and more seriously. My mother is having a fit. She's scared to death my daughter will get seriously injured or worse. Unfortunately with riding and horses, danger is part of the equation. You can be as safe as you can and take every precaution, but even the most predictable bomb proof horse can have a spook or a bad day.
As far as knowing when your child is ready, if you have a trainer, your trainer will probably be able to have a pretty good judgement on when she's ready. If you're teaching her, and you feel she's ready and your daughter feels she's confident and ready, then there's only one way to find out. Best of luck to you and your daughter and mom!