Does a capacitor respond faster than a battery?
I don't know if a capacitor is faster but if I had to guess, I would guess that the electrostatic mechanism in a capacitor is a lot faster than the chemical mechanism in a battery.
But even if batteries were just as fast as capacitors, you would still use capacitors. One thing the capacitor has is a lot less inductance than the battery. Furthermore, there are usually a lot more components that need a low inductance power source in a circuit there are batteries, and it is a lot easier to place a capacitor right next to each component (thus minimizing wiring/trace inductance) than it is to give each one its own battery.