Why do we short the input when calculating the output impedance?
The two test configurations tell you two different things.

If you short the input, you're testing the output impedance of the circuit as it is normally used. This is because you typically drive the circuit with a (relatively) low impedance source.

If you open the input, you get the \$Z_{22}\$ as defined in circuit theory as a characteristic of the circuit itself (without effects from any driving circuit).

The first option is often more practically useful.