How do capacitors work
Its correct..... Capacitor can discharge very rapidly thus able to deliver higher output power. However, their input power could be very less depends on the current capability of the power supply that charges it.

If look closely at xenon flash lamp in cameras, most cameras operates on tiny battery but xenon tube operated from them flashes at kilowatts. This is true because a high voltage capacitor inside the unit is charged to +350V by a boost converter. The capacitor takes few seconds to be fully charged. During flashing the entire charge is dumped into xenon tube in mere milliseconds.

Power is actually the rate of energy conversion. Capacitors can do this very fast thus can exhibit very high output power.

This is why capacitors are also known as condensers or accumulators.