Atkins is LCHF and it was never a quick fix diet, it was always a lifelong Way of Eating, referred to as WOE. Dr Atkins took his patients through 4 phases, Induction, to eliminate toxins and 're-boot the system', for most people recommended for 2 weeks only, then ongoing weight loss, where you increased daily carbs by 5 g, for a week at a time, until you stopped losing, then drop back 5g - that was to establish your personal carbohydrate tolerance, then pre-maintenance then maintenance, the objective being to hold your reduced weight for ever.
The Atkins Corporation is now basically a money making convenience food company, but Phinney, Voleck and Westerman, who were commissioned to write the latest book, based on lots of recent science, are foremost experts in the field.