Training fasted: just about weight loss?
Advice in book Racing Weight by Matt Fitzgerald goes something like this, paraphrased:

Yes, training without carbohydrates will train your body to use stored fat better. But your capacity to train will go down (not enough fuel!), and net result will be less improvement.

It references this study, which compares two groups of athletes on hi-carb and low-carb diet during a hard training block. 

One of the general themes of the book is, you want to improve your performance and body composition, not your weight per se.