The use of water for moderation is one thing, the use of water for cooling is another thing, even if in many reactors water is used for both purposes.
A reactor can be moderated with something else than water, e.g. graphite, but it may still need water for cooling.
The amount of water needed for cooling is much more than needed for moderation.
So there is no doubt that many "non-water moderated reactor designs" still need copious amounts of cooling water.
Any "non-water moderated reactor design" that does not have liquid fuel, i.e. it is not a molten-salt design, must have a cooling fluid, though the fluid in the primary cooling circuit may be not water, but something else, e.g. molten metal (e.g. molten sodium) or supercritical carbon dioxide.
I believe the point was that non-water moderated designs typically operate at higher core temperature than LWRs, so they can reject waste heat at higher temperature (or reject less waste heat per unit of electrical energy produced), and that makes rejection to air more practical.
A very high temperature reactor might even be able to work with an open air Brayton cycle system, which would allow heat to be directly exhausted in that air stream. It would probably still need an in intermediate heat exchanger so the air wasn't being irradiated with neutrons.
No, because it would require very high temperature. The air coming out of the compressor of a gas turbine would already be hotter than the water/steam coming out of a LWR. It would likely involve a core temperature of around 1000 C. The turbine inlet temperature of a modern jet engine is at least 1500 C. There would likely be thermal NOx production, so post treatment of the gas might be necessary in an open cycle system.
A reactor can be moderated with something else than water, e.g. graphite, but it may still need water for cooling.
The amount of water needed for cooling is much more than needed for moderation.
So there is no doubt that many "non-water moderated reactor designs" still need copious amounts of cooling water.
Any "non-water moderated reactor design" that does not have liquid fuel, i.e. it is not a molten-salt design, must have a cooling fluid, though the fluid in the primary cooling circuit may be not water, but something else, e.g. molten metal (e.g. molten sodium) or supercritical carbon dioxide.