How would a more efficient heat exchanger reduce the resulting temperatures of the river water used to dumps the heat? A river full of dead fish is even worse PR than long term radiation danger.

This is specifically a problem with building a nuclear power plant on too small a body of water. The ocean doesn’t have this problem. Neither would the Great Lakes.

Actually it's a problem that arises when you build a nuclear plant next to an intermediate body of water. When the river is too small (or irregular, like the Loire) you build cooling towers on your plant, and you are fine (or at least, you need much more extreme weather conditions to have issues). The problem with plants built on the Rhône is that there is usually enough water in the river for direct cooling, so the designers didn't bother with a cooling tower[1]. But when the water flow is low, you need to stop the plant.

[1]not all of them though, in some cases the Rhône's flow wasn't enough for direct cooling and they went with cooling towers: The Cruas-Meysse plant has cooling towers and it's not affected by the current heatwave. Also, the Bugey and Tricastin plants have a mix of both (2 reactors with direct cooling and two with cooling towers) and only the reactors with direct cooling are affected.

I thought the cooling towers at Tricastin were only for the fuel enrichment part

The problem is regulations. If the river is already hotter than the max temperature allowed as output, then it’s not possible to operate, for regulatory reasons. Temperature delta is on the order of 0.1°C