You don't need to eat carbs to need insulin. The entire principle behind modern insulin therapy is to have a steady, small stream of insulin to cover the output from your liver, which continuously releases glucose throughout the day (noteably as a result of gluconeogenesis). Therefore a T1 will always need to take some form of insulin regardless of how may carbs they eat. It's worth noting that the standard procedure for basal testing (no carbs for an entire day) is designed to work out the insulin need to meet an individual's basal requirements.
If someone is able to cut down their carbs to the point that they don't need insulin, they are either a)not actually a T1, b)a T1 but with some residual insulin production (which raises a lot of questions over whether we actually regenerate beta cells and if so, opens up an opportunity for a cure) or c)not actually controlling their diabetes. Furthermore, I suspect even in this situation, the need for insulin would come back. Insulin prevents fat metabolism by preventing the release of glucagon. No insulin means more glucagon, which releases glycogen from the liver, which becomes blood glucose - which will require insulin for metabolism. I can't quite recall whether ketones can be metabolised by the brain but you'll note that if the liver is still releasing glucose, in any case, there is glucose available to fuel the brain.
So the point still stands - you don't actually need dietary carbohydrate. Again, this isn't to say you should never eat bread or pasta and live entirely on eggs and avocados - just that you don't need to base all your meals around a starch. In fact, if you're overweight, it's better NOT to base meals on starch. Again, insulin prevents fat metabolism and encourages the conversion of blood glucose to fat. If you're looking to lose weight, you probably need to reduce your insulin intake, and that'll mean modifying your diet to ensure you retain blood sugar control. The entire 'wisdom' behind the supposed inevitability that T1s always get a bit fat is entirely because the medical authorities work on the principle that your adjustments should always be on medication rather than diet.