But they charge up relatively quickly.....................
Discharge rate is hi - but as one post here noted the voltage drops. The other issue is the time constant for either discharge or charge. Usually the time contact is very small because the total capacitance is low. Ultra capacitors or this tech will have bigger time constants which is function of the total capacitance. Anything that rivals the energy density of a battery will take significant time to charge.
Also, at some point you will be pushing the limits of a common 15A outlet.