A freshly baked cookie may be too fragile to lift even when it has finished baking. A short rest on the tray can allow it to firm before moving to a cooling rack.
Use the recipe's signs of doneness, which may include set edges and a softer center. Do not keep baking solely because the whole cookie is not crisp while hot; that can produce a dry result after cooling.
At the same time, cooling is not a substitute for cooking raw dough properly. If the center is visibly raw, check the bake rather than assuming it will sort itself out on the counter.
Move cookies with a broad suitable spatula once they can hold together. Cool fully before storing in an airtight container, unless the recipe advises otherwise. Trapping warmth and steam can soften a crisp batch you worked carefully to achieve.