Yes, bleach takes color out, but the remaining color (because rarely does it actually take it completely out, as per your reference- note that there is no white in that image at all) will be a paler shade of whatever the original dye color was. On denim, the indigo leaves a beautiful sky blue. On nearly any other dark dye color you can get icky browns, dull grays, and even orange tinged barfy colors. In addition, bleach is very harsh to many fibers, it can actually burn a hole right through fabric.
In this specific case, snow/ frost is what effect is needed. Fresh snow is pristine pure white.