Is it really important to cut down the Carbs to get a lean body?

Not necessarily, but it also depends on what you mean by "lean body." Lean means lacking fat. As Joe noted, long distance runners (who burn hundreds if not thousands of calories) are very lean and also aren't aren't very muscular. The body used glucose/glycogen or ketones as fuel. All carbs are broken down in the body into glucose. If your body uses up the fuel you eat in the form of carbs, you will not store fat. If you eat more carbs (or calories in general) than your body uses, your body converts that extra fuel into fat and stores it.

Most people who are overweight are not long-distance runners. They are not burning the food they consume. A low-carb diet can allow a person to eat more fat, calories and protein and force the body to burn ketones from stored fat as fuel. That is why a keto diet is popular -- and effective - for a lot of people.

It is important to cut fat.

Insulin is the fat storing hormone. If you want to lose weight you need to be insulin sensitive. My insulin sensitivity was in the toilet...I had severe insulin resistance resistance with chronically high insulin. My cause was processed sugar.

"#3: Optimize your carb intake.
If you’re sedentary and overweight, optimizing carb intake may mean to eat a very low-carb (less than 50 grams a day kind of low) diet. For others it means to restrict carbs to 100 to 200 grams a day, or try carb cycling. For recreational trainees, it might mean to eat moderate to high-carb on training days and low-carb on off days."*

*25 Simple Ways to Improve Insulin Sensitivity:
http://main.poliquingroup.com/ArticlesMultimedia/Articles/Article/1269/25_Simple_Ways_to_Improve_Insulin_Sensitivity_&_PREVENT_Diabetes.aspx

most thin long distance runners are eating lots of carbs.

it will really help cutting the carbs