Dry cooling tower exists but they are more expensive. Natural cooling with the evaporation eventually runs into the problem due to built-up of salts which forces you to replace non-evaporated cooling water with fresh river water and lets you dump the heated water into the river (which is limiting factor due to environmental concerns). The less water is flowing and the higher the temperature of the river are the sooner this point is reached

My hunch is evaporative cooling uses not very much water. You could also fill a small dam when the water flow is higher. All means higher cost, of course.

It generally helps to no use hunches when it comes to hard technical questions.

Ok, the hard technical answer is that a phase change from liquid to gas eats a lot more energy than just heating the liquid however many degrees. :-)

My back of the envelope calculations say it's about 10 times less water to evaporate.

There are detailed diagrams and formulas to calculate that. Engineering, not back of the envelope highschool physics.

But guess what, people designing an building power plants know this. And whatever is built is the best compromise possible at the time. Backnof envelope calculus in 2023 or not.