It's great that when a child is so capable at academics. That means she can relax on that front and have more energy to work on other issues.... like, risking failure, tolerating being the centre of attention, being willing to assert herself. Something like Rainbows could help a lot. Also, being so bright herself, she may not be tolerant of others who are less able, it's another skill she may need to work on.
I suggest.... on The fear of failure thing.. you could model the best response yourself. Try to do things that are difficult, let yourself be seen to fail or struggle, bumble a bit but keep trying anyway without being upset at slow progress. It would be very helpful if you and she do some things that are fun but you aren't much good at (like ....playing tennis?). Something where you can both laugh about being rubbish at it but keep doing it anyway. Not everything in life is about progress or success.
I don't know why you're bothered about her not being pushed at school on the academic front, it sounds like she has heaps of other things to work on. Once she knows that it's okay to fail or be mediocre, her confidence will come on in leaps and bounds, she will come on so much better in everything else. Only then will she start to feel more confident about speaking out in class, etc., and really be able to flourish.
DD was a bright but silent mouse at school until they moved her to another class at start of Y2, specifically to put her with another very bright child but also because she was dominated by some in her social circle, had become too emotionally dependent on them. When she made new friends in the new class, she took off emotionally and intellectually. One of her new y2 best friends is not esp. bright, but has the best possible attitude about failure, rarely bothered by it. Has been enormously positive example for DD.