Carbs raise blood glucose ,, insulin is released to lower blood glucose and it does so by storing glucose in BOTH fat cells and muscle cells (as glycogen). So no insulin is not just a fat storage hormone (why do yiu think body builders take insulin , to get fat ? No. )
Can you cut carbs out of your diet and deplete glucose and glycogen stores and thus no longer store those as fat and instead start actually using fat itself as the energy source ,, sure. BUT if you consume MORE calories then you need your body will begin using protein and converting that to glucose and you'll still get an insulin response even without carbs ,, so you'd still need to reduce calories to burn fat