As I pointed out further down in my comment, solids and liquids have a much higher heat capacity than air (or in other words, they can absorb a lot more heat before they warm up), so most of the heat dissipated to the river would end up stored in the Earth’s Crust and Oceans and very little of it in the Air.
Even if the water is used from a river, the heat still gets dissipated into the air from the surface of that river.
So river cooling is still just air cooling with extra steps.
It all makes its way into the cold vacuum of space eventually
As I pointed out further down in my comment, solids and liquids have a much higher heat capacity than air (or in other words, they can absorb a lot more heat before they warm up), so most of the heat dissipated to the river would end up stored in the Earth’s Crust and Oceans and very little of it in the Air.