Can anyone explain why the popcorn graphine acting as the anode and cathode would increase capacity and speed of charging, but not speed of discharge?
Surface area, conductivity of the electrodes, and the ability to reject heat during charge and discharge cycles.
Cell charge and discharge rates are limited by their internal resistance, which can cause too much heat to build up within the cell, limting efficiency, and causing cell damage and failure.
That is bad with a lead-acid cell. It’s really, really, really bad with a li-ion cell.
The circuitry being fed determines how much draw there is. You have a number of outlets running from the same circuit breaker but the actual draw is determined by the number/type of devices plugged in.
Nothing wrong with a faster discharge. It provides more amps for torque sensitive equipment.
Discharge rate is generally gated by the impedance of the device ("load") drawing current from the battery -- not the battery itself.