This seems completely wrong.
The justification is given by V=IR. Assuming current is finite you would indeed conclude that V=0. But why would you assume finite current?
Yes, real-world currents must be finite, but real-world resistances must be nonzero. This is an idealization; the idealized values don't have to be physically attainable.
And, in a real-world approximation of an ideal short circuit, one sees very large current; nonzero voltage, infinite current, and infinite power seems like a much more accurate idealization than the finite current, zero voltage, zero power idealization.
Thus my question. Is this idealization of finite current and zero voltage really the common one to make? And why?
Edit: to make it explicitly clear, in this idealization, the parameters of the ideal circuit are allowed to attain idealized values — specifically, a priori, a literally infinite for current is allowed (for mathematical precision, I mean the extended real number ∞). With R=0 and I=∞, Ohm's law puts no constraints on the voltage; every extended real number value for V is consistent.