Its only where shortcuts have been taken with cooling that there will be issues.
So, everywhere, in a Capitalistic society
That seems like a huge disadvantage for nuclear power if you need to guarantee strict ideological commitment to ignore cost-competitiveness not just now but into the future. Renewables and batteries are not only cheaper now but have far less scope for major failures based on some MBA cutting corners in a decade.
You think big battery plants can't have issues with cost cutting?
So basically no.
What do you consider adequate cooling? Remember that for every GW of electricity you need to remove 2-3 GW of heat. That's a lot of heat. Strapping a giant PC radiator fan to the plant won't cut it.