It is a question of timescales. Capacitors are typically used to “smooth out” power over timescales of nanoseconds to seconds (super capacitors can sometimes go up to minutes or hours). Batteries can smooth out power over timescales of minutes to years.

Batteries are also better at “clamping” to a certain voltage, while capacitor voltage is typically more variable.

Yes batteries can do this for a DC system like in your car. But that is mainly a voltage regulation function.

Power quality correction is a consideration in large industrial AC power networks. It helps with the problems that arise from running large amounts of electric motors where the power factor gets out of balance between inductive and reactive currents.

Since electricity essentially sees the coils within a motor as an inductor, it affects multi-phase power in a way that's detrimental to its ability to transfer power

So capacitor Bank systems are designed to correct the issue so that the overall AC power quality isn't effected for downstream users.

The actual regulation in a car is done by the ECU or some circuit inside the alternator that limits the output voltage to a certain amount.

The battery is mostly there so that when there is a load transient, the regulator mentioned above can have a relaxed response time--the battery takes up the slack when there's a sudden load increase or decrease.

If a control system has to react quickly, it's more liable to instability like oscillation and divergence. Better to keep it safe by making it slow to react, especially if you already have a big ass battery that can smooth things out that you need to start the car.