I think your post wins the "worst analogy in this discussion" award.
There is so much pseudo-science on every side of this debate that I'm against just about anybody who takes an absolutist position in the way you have.
Personally, from my own research and experimentation in my own diet, I firmly believe that weight is an I/O issue though good health is more complex than that.
While Atkins never specifically advocated for calorie restriction, it strikes me as common sense that some portion of new Atkins dieters experience positive results due to calorie restriction. It can be difficult to replace the number of calories in pastas, pizza, etc, in part just based on caloric density.
It's also confused by the fact that Atkins and other (if not "fad" then "theme" might be a better word?) diets have to change both physical, calorie I/O issues, and mental issues to truly be effective long term. While I truly believe that for weight loss, 2000 calories of fast food is the same as 2000 calories of lean protein, fruits and veg, after the period of strict control (the "diet") the person eating 2000 calories of fast food has not done much to truly change their habits and mindset.
I, personally, have accomplished a lot by intermittent fasting (didn't know it had a name until later, to me it was just about not eating as much during the day in favor of a bigger dinner) and learning that it's OK to feel not-full. And to learn the difference between hunger and feeling not-full. I went from 210 to 170 and lost 55 pounds!! of fat during the process. I have a big dinner (usually 50% of my daily caloric intake) and dessert every night (another 20%). That's complimented by a lunch of around 25% of my daily intake. Very infrequent snacking. No breakfast, even on Gym days, and no soft drinks only water, aside from alcoholic beverages.