The air from my air conditioner sits on top of me like a blanket, not touching the parts of me in contact with the mattress. If the mattress itself is actually cool, it will stop me from sweating. The recycled air thing is just personal preference, breathing fresh air feels subjectively better than recycled air so I keep my window open as much as possible. If the weather is hot, but my mattress is cold, then I can keep the window open and still be cool on hot days.

I can't speak to the person you're replying to, but it's like the difference between an electric blanket and a space heater. It's energetically cheaper to cool just the bed than the whole room, and it won't dry out the air as much.