There are many questions about capacitor discharge rise time here on the electrical engineering board but all involve transistors or are oscillating circuits rather than a purely shorted out capacitor. The capacitor discharging equation assumes that current at time zero is at peak. However, I find it difficult to believe that the current rise time is effectively infinitesimal in real capacitors. Does the current of an RC circuit truly peak at time zero?
Real capacitors have some amount of parasitic series inductance that limits the rise time of the discharge current. Real capacitors also have equivalent series resistance (ESR). This make every capacitor circuit using real capacitors an RLC circuit with small values of L and R.