It's not a question of enough cooling water. It's a problem of how much the plant is allowed to heat the river. Because apparently keeping a stretch of river below 28*C is more important than climate change.
Not killing all the fish in a river is also generally something that should be aimed for, since that can have knock on effects on ecosystems. We’d rather not find out what happens and accidentally instigate some kind of environmental disaster that might lead to farm failures or something unexpected.
The limit on the temperature is as low as 26C or 79F. It's not like the river is boiling. Furthermore this heat dissipates downstream, it's not affecting the entire waterway. This heat has the potential to harm fish in a segment of a few rivers. By comparison, throttling carbon-free energy production has the certain impact of producing more greenhouse gases, which will contribute to global climate change. I seriously don't see how someone makes this kind of tradeoff - it's like the same mentality that held up a solar farm to relocate tortoises.
* many rivers have some kind of migratory fish integral to ecosystems in both the river and whatever the outlet is, so you can’t just mark rivers as discrete segments and say the effect is isolated
* the oceans haven’t warmed by all that much and we are already seeing the impacts, it’s not really that much of a stretch to say rivers would see similar effects
* given humanity’s track record with the question “how bad could this be?” and unintended consequences, erring on the side of caution seems warranted. In particular, humanity has a bad track record with river management
Remember, though, the consequence of shutting the reactors down is an increase in global climate change. The consequence of running the reactors are increased river temperatures in a handful of rivers - which as you point out are also impacted by climate change!