These aren't independent variables. The point is that a half an hour run only burns around 300-400 calories, a deficit which could more easily be accomplished by diet. Also, it's possible that the run would increase your appetite.
The point isn't that exercise _can't_ work, it's that aiming for dietary changes has a higher average ROI than exercise.
But it's more complicated than that. Genetics cause one's predisposition as to how responsive a person is to cardio; see the University of Bath study and/or Michael Mosley's "The Truth About Exercise".
>but the overruling factor in weight loss/gain is the ratio of work (exercise) to energy input (calories).
I don't think you are off, but the key is not weight loss/gain...the key is healthy weight loss/gain and that has everything to do with what you consume.
I don't know what a normal low fat diet is, but the overruling factor in weight loss/gain is the ratio of work (exercise) to energy input (calories).