Everyone has different views on parenting. I think you can't over shield as a child needs to be able to build up resistance so playing in the mud etc is normal. As long as the parent is thinking and having the child's best interest at heart no one is really wrong. Everyone has to be different so I have never heard of a child not having any sugar and you may be totally correct. You must give her a substitute like honey then? I think the dietary requirements and the parent worrying about it has changed a lot from when I had my kids over 20 years ago so am a little clueless right now.
It is because of her allergies that it has to be this way. She eats some meats and vegetables and wild berries. I call it the baby cave girl diet. She gets sugar from the vegetables and berries she eats so there is plenty for her brain to develop and her energy levels are constant and high due to the amounts of fats. There is actually more energy in fats than sugars and without sugar present, the fats pass through and are healthy. Our bodies aren't designed for constant high sugar levels.
I know there are lots of sugar in fruits and berries. I suppose you have to be very careful especially with allergies. it becomes a way of life and has to be that way. Maybe she will grow out of them as she gets older. My son is lactose intolerant and milk affects him.