You absolutely can experiment with substituting applesauce for some of the fat in traditional recipes for baked goods. But fats play various important roles in baking, and messing with them too severely can cause problems.
One of the most important tasks played by the fat is keeping the flour proteins from mixing with the moisture in a batter and forming long strands of gluten that will make cakes and cookies tough. Sugar can play that role effectively, too - and, of course, the applesauce you add contains some sugar - but loading your recipe with sugar would allow you to cut the fat, but not necessarily the calories.
To help cut down on the formation of gluten, mix the liquid ingredients and sugar well (using an electric mixer if practical to beat as much air into it as possible). Mix the dry ingredients in a separate bowl, and don’t combine the two until the last moment. Then mix them gently by hand, and as little as possible. Gluten begins to form as soon as the flour comes in contact with a liquid and gets more active the more you stir it.
Fat plays a role in determining the shape and texture of a finished cookie - butter in cookies causes them to spread, producing crispness; cookies with vegetable shortening do not spread, and are more cake-like. You won’t be able to produce a crisp cookie with a fruit puree.
Finally, fat has a lot to do with carrying the flavor and nutrients in baking, so you should not do away with it entirely, or you’ll just lose a lot of the flavor.
There are commercial fruit puree-fat replacement products on the market. Smuckers and Sunkist each make them. In the case of Sunkist, the product includes a mix of pureed plum and apple as well as pectin (a natural thickener), dextrose (a natural sweetener), and maltodextrin (a complex carbohydrate made from corn starch and used as a fat replacer). Basically, it’s a jam, but augmented to help produce certain results. In the case of these products, the instructions specify adding half as much puree as the amount of fat called for (so 1/2 cup of puree instead of 1 cup of oil).
You might want to use that as a guide. Why not use a half a cup of applesauce and 2 tablespoons of oil in your cookie recipe and see how it turns out? That’s almost 1,600 calories fewer than the recipe started with. You might have more success (and be more satisfied) with cakes and muffins made with fruit purees, but starting with cookies is a daring move and you may surprise us all.