I think we lose weight on Atkins diet for two reasons:

(1) When we are in the ketosis state we use ketones for fuel, we waste some of them and we cannot make new body fat.

(2) We eat very small amount of carbohydrates so the the insulin overshooting effect which causes craving is not likely to happen. So we have better control over our appetite.

Low GI food can help with item (2) only. For item (1), if we get enough glucose to use as a fuel regardless of how fast or slow it comes, we will not need to produce ketones and will burn fat the old way which has caused us to be overweight.