When I started reading this, particularly the first line quoted above, I was ready to call BS on it, and almost stopped reading. The idea that that the river being a bit warmer would make it ineffective at cooling the reactor was too silly. But then I got to the second quote and realized they just don't want the reactor to make the river any warmer than it already is.

The Wikipedia article for Bugey Nuclear Power Plant says "Some of the cooling comes from direct use of the Rhône water (units 2 and 3) while some is done by the use of cooling towers (units 4 and 5)." Cooling towers use evaporation to remove the heat, and water has high enthalpy of vaporization so they'll probably still work even with warmer input water, but if the direct cooling system is not designed to boil the water then it might fail if the input water is too warm.

https://en.wikipedia.org/wiki/Bugey_Nuclear_Power_Plant

I live near https://en.wikipedia.org/wiki/Oyster_Creek_Nuclear_Generatin.... When the plant was running, it would suck in so much water that one river would run backwards, pulling in brackish water from the bay (near the outlet of another river), and depositing the outflow into a different river, which went back into the bay about a mile away. Both the inlet side and outlet side were rich fishing and crabbing grounds. Now that the plant is shut down, both the inlet and outlet are nearly stagnant, and all the seafood is gone. The outlet was a particularly popular fishing spot, because the warmer water both attracted more fish and greatly extended the fishing season (even through the winter, when you couldn't catch much elsewhere).

My thought is that hot and humid air is holding more moisture as it is, so it’s harder to evaporate more into it.

If they’re drawing from the river to evaporate, and flows are sluggish (because of evaporation upstream or lack of precipitation), they also have to draw less when they need more.