Thank you for your thoughts and question. It is very rare to see questions here on HZ, there are a lot of opinions though, some of them can be very strong and even offensive.

To answer your question: The crux of the talking point or “theory” does center around the energy released by nuclear fission, and not necessarily the “heat” caused by that activity.
The heat is simply the moving of molecules that have been accelerated by the kinetic energy obtained from radiation and fast moving particles that are ejected from the nucleus of an atom. Those fast moving particles approach the speed of light, as your link to the “blue light” points out.
It is the PROCESS and the AFTERMATH of that process that I am trying to explain.

We could compare the fission of atoms to triggering a shotgun, the shot flies off at high speed, although only reaching a small fraction of the speed of light.
We could use the chemical energy and the flying shot from many shotguns to heat a room that could heat water that in turn would turn turbines to generate electricity. We could make the room walls heavy enough to withstand the heat and the damage that the flying shot would cause. If we did generate energy in such a way it would be far more benign than the nuclear process. The speeding shot from the shotgun will only cause the molecules of the shot and the walls, to increase their activity, once such extra thermal energy is radiated away the molecules return to their original energy level. Such is not the case in the nuclear process, the produced radioactivity “heats” the environment for thousands of years.
I am curious about your statement of: “Even if they all simultaneously had a meltdown, it wouldn’t raise global, let alone local temperatures a lick.” Did you actually figure out the amount of energy that would be released in such an event and how long the “produced” radioactive atoms would be “heating” the atmosphere?

The information, from the link that you gave, is that there are 436 active nuclear power plants, producing 370,494,000,000 watt hours of electricity, remember that this quantity is only a small fraction of the total energy that is being created, even if this quantity of energy was created by burning fossil fuel, it still would represent only about 30% of the total energy that was being produced, the rest is wasted heat. When fossil fuel is used to produce electricity, the fuel is depleted and the resulting heat is in the form of molecular activity, the molecular activity and atomic excitement is raised to the level where infrared energy is released. When the fuel supply terminates, the infrared and molecular excitement also cease within a relatively short time.
The nuclear reaction is different, once you set that activity into motion it does not terminate for many thousands of years, and during that time it “heats” the surroundings.

You can get very detailed information about nuclear reactors from this site.

http://www.nr.titech.ac.jp/coe21/eng/events/NuclReactorTheoryTextbook.pdf

The rest of my explanations dwell on the extrapolation of what will happen IF the Earth’s atmosphere is “heated” for a long time with high energy nuclear activity.
Basically the changes would consist of the following:

If the atmosphere gets hotter it expands. (obvious physics)
If the upper atmosphere gets very hot, the molecules will escape the Earth’s gravitational field. (thermosphere area)
If atmospheric molecules escape into space, then the atmospheric pressure will decrease.
If the atmospheric pressure decreases, then there will be a variety of changes in the global environment.
The severity of the changes will depend on the severity of the atmospheric loss.