Studies have shown that simply seeing sweet foods (without even eating them) can cause your body to release insulin. This abstract of a review paper includes the sentence:
high acute levels of insulin can be produced by simply seeing and thinking about food and that individuals showing this response show a greater tendency toward weight gain in a food-abundant environment.
Therefore it stands to reason that eating sugar free sweet foods can have the same effect. Whether this then leads to you getting more energy from the other food you have eaten I do not know, but the abstract covers other studies that directly connect these reactions to weight gain due to the next thing you eat.
I long ago observed for myself that if I had sweetener in my early-morning coffee, when the coffee cart came around (a thing that used to happen in offices in the 80s) I was unable to resist having a bun or muffin with my mid-morning coffee. But if I had a teaspoon of sugar in that morning coffee, I could last till lunch with no buns or muffins. The net savings of sugar and fat was definitely on the side of sugar in my coffee. Your physiology may vary, but science says my observation was real.
However, your summary "the body thinks they are sugar" is probably not supported. Rather, the mind thinks they are sugar and things go from there.