HZ123 you are talking about the 'Steering depressurisation/bypass pin' the original question is about the Nose gear downlock pin...which prevents the gear from retracting.
Most airlines I have been involved with fit the downlock pin only for towing the aircraft, often they are towed without any hydraulic pwer on (towbarless tugs etc). For normal pushbacks then the downlock pin is not fitted. However it will depend on each companies own policy. I do know of one major airline at LHR that fits the downlock pins for push backs. just means that the ground crew will have to show 2 pins to the crew before they taxi off.