True but for that split the cap would just be able to switch on the
transistors or FETS and be barley noticeable.

BUT>Think of a REAL CAR with a great big whopping sound system restricted to a 6 to 9 plate lead acid battery, every time the amplifier or amplifiers require a huge lump of current draw (eg. that pain in the ass doof doof sound you hear from miles away!)
the battery suffers to supply the correct amount of current causing the driver ic's in the amp to run uneffeciantly, over heat, shutdown or in some cases 'crap out all together!'

BUT by placing a massive power capacitor
parrallel across the +tve & -tve terminals of the battery, smooths and provides a more efficiant supply current and voltage(only for short bursts) hopefully correcting any supply problems alltogether.
Its a similar principle in rc.
Hope it made sense.