You don't explain what your reasons are for only putting one child on this regimen. Does he have a particular problem with cravings, weight or tooth decay? What about behavior/hyperactivity? I think it's important to identify the goals and perhaps we could be more helpful.
That said, it's much easier if EVERYONE follows the same health regimen, rather than singling out one person and saying, "You can't, but everyone else can." Kids sneak, crave and act out. So you want to get off processed foods entirely - go to real cooking. Read labels - if there's a long ingredient panel, it's processed. Look for sugar or corn syrup or high fructose corn syrup in the ingredient list - you will be shocked at how overloaded things are. I tried to buy a loaf of bread last week - the "healthy" and "whole grain" breads were loaded with high fructose corn syrup!
Absolutely do not buy "sugar free" or "low sugar" products if they contain aspartame, sucralose or saccharine - those are the chemicals in things like Equal and Sweet & Low and whatever the 3rd popular one is (yellow packet - name escapes me at the moment). These do nothing to cut the sugar cravings or help weight loss, and they are probably dangerous.
There is a natural product called stevia - plant based with no calories. You have to be careful about where it's made. I buy mine on line from a reputable food science company with good FDA certification that does its own manufacturing here in the US. Rebiana is also safe - it's actually from the leaf of the stevia plant, which is the sweetest part so you can use less.
Natural sugars occur in a lot of foods (e.g. fructose in fruit) so your family can get plenty of sweet tastes from natural sources. I'm not sure why you ask if it's healthy to avoid sugar. Refined sugar does nothing for anyone's health, but avoiding natural sources like fruits and vegetables is a terrible idea. There are also certain supplements that make use of plain fructose which are not only safe, they are beneficial when limited and still part of a comprehensive supplement.
In general, if you substitute the darker versions of white foods, you'll do better. Sweet potatoes, oven-fried or grilled, are a great substitute for white potatoes. Brown rice and whole wheat pasta vs. the white versions, whole wheat flour vs. white flour, and even brown sugar or unrefined sugar vs. refined cane sugar are all important adjustments to make. Just taking out "sugar" and leaving in all these simple carbs that convert immediately to sugar will be counterproductive.
If your child has behavioral issues, they can be much improved with the proper products. If he has a weight problem, the way to address it is not just eliminating one element - it's to nourish him properly which the AMA says must include supplementation. If you need more explanation or info, let me know.