They had to be shut off during heatwaves mainly to limit the raise in temperature of nearby rivers, due to the river heating, the limited flow of the rivers, and the raise in the reactor's egress water temperature. It is not clear if the raise in the egress water temperature had a big impact, it seems more likely that the heatwave made the rivers less able to receive the excess heat. The cooling systems were still fully able to cool the reactor, but not without shedding a lot of heat to the rivers (as usual).

Of course, this is not at all contrarian to your point, I made the precision because "no longer could be cooled adequately" could be misunderstood.

They were also some problems with the efficiency and reliability of backup generators during heatwaves, but this is not very hard to handle (the backup generators are much easier to cool than the reactor itself).