Because the AC transformer is just providing enough current to run the control box and trickle charge your battery. The motor has a higher amperage draw than the transformer can handle, especially when it first starts operating, which is why you need the battery. It's just like how in your car, you only need the battery to start the car, you don't need it to run the car because the alternator takes care of that for you.
Why do I need the battery if I have the electrical outlet?
Because the AC transformer is just providing enough current to run the control box and trickle charge your battery. The motor has a higher amperage draw than the transformer can handle, especially when it first starts operating, which is why you need the battery. It's just like how in your car, you only need the battery to start the car, you don't need it to run the car because the alternator takes care of that for you.
Maybe 2 years max