Mix flour and confectioners' sugar together in a bowl. Cut butter into flour mixture using a pastry blender or 2 knives until mixture resembles coarse crumbs; press into a 9x13-inch baking dish.
Bake in the preheated oven until crust is lightly browned, 20 to 25 minutes. Remove from oven and cool completely.
Beat cream cheese, 1/2 cup white sugar, and vanilla extract together in a bowl using an electric mixer until smooth and creamy. Fold whipped topping into cream cheese mixture and pour over crust. Refrigerate mixture until cream cheese filling is chilled, about 30 minutes.
Pour cold water into a bowl and sprinkle with gelatin; stir well. Pour hot water over gelatin mixture and stir until gelatin is dissolved. Stir 1/2 cup sugar and salt into gelatin mixture until sugar is dissolved; add lemon juice and cool to room temperature. Fold mango into gelatin mixture and refrigerate until gelatin begins to set, 15 to 20 minutes.
Pour mango gelatin over cream cheese filling and refrigerate until completely set, about 1 hour.