Does the water cooling system has to go through the welder? does it serve any purpose of cooling component in the welder?

Does the shielded gas have to go through the welder? my guess is it would save argon gas with an on/off switch controlled by the pedal - on only when welder is in use. Is this a correct assumption.

Any help/answer is greatly appreciated and thanks in advance, Just learning and welding for a hobby.