Agreed for the most part. I think there is a synergistic effect of adding exercise to a diet and there are multiple non body fat loss related benefits to exercise. And that doesn't necessarily mean running marathons or signing up for cross-fit. There's a ton of evidence that's been coming out which shows the most efficient form of exercise is plain old walking. Just walking for 45 minutes a day provides a ton of benefits in addition to burning calories. But, your initial point that cutting carbs and sugars is the key. Minimizing starchy carbs (white bread, white rice, potatoes, pasta, etc) and cutting way down on sugar is essential. Don't drink any liquid calories if you are trying to lose body fat. Fitness/weight loss is: 60% diet, 30% exercise and the digestive process itself accounts for the other 10%
I put over 1000 miles on my bike one summer — lost 15 lbs. I rode everywhere. Limited carb intake two years ago and lost 52 lbs.
No doubt I was in very good shape that summer. But my goal was more to lose weight than to get into shape. Plus, carrying that extra weight while exercising is hard on the body. Find a low impact exercise.