First, I don't do Atkins FTR.
The study was pretty long term. The people in the Atkins group lost far more than glycogen. And insulin resistive people lost like 2 or 3 times the weight of other diets so even with glycogen loss, they lost substantially more.
Also, what does insulin sensitivity matter in the absence of carbs? Especially if eating some carbs improves that sensitivity (when it would matter what your sensitivity is). If I remember the Atkins group increased sensitivity anyway.
The triglyceride thing didn't pan out in that study. All blood markers in the Atkins group improved and every single improvement was better than any other diet in every measurement.
And also remember the atkins one was the only diet that was not semi-starvation which IMO makes every single other diet impossible to sustain over time (there are plenty of studies showing this). Which is important to note because exactly following the diet was not studied, what people actually did when "following" the diet was measured.