It is pretty funny how parts of the debate never seem to be resolved and some are the simplest parts until you think about what the misunderstanding really is, idealism versus realism.
The analogy of a black body versus a gray body for example. A black body is an ideal representation of a "perfect" radiant source and since there is no such thing outside of a lab, a gray body is supposed to represent real world conditions, but it isn't something that can be observed in nature either. You can call something a gray body or black body, but there are still arguments in physics about what each might be on a large scale.
In a lab, a black body is a tiny slit in the shell of some volume at constant temperature. The interior of the volume is coated with something like lamp black and for precise measurements there might be a tiny object opposite of the slit to provide a stable energy source at the same temperature as the volume. Since different frequencies of electromagnetic energy have different wave lengths, measuring a wavelength might require adjusting the thickness of the shell so that random, isotropic, radiation appears to be polarized for the instrument used to measure the energy. Since this slit in the shell of assumed negligible thickness is in "equilibrium", it has a balancing radiant energy flow into the slit equal to the flow out of the slit. So if you are measuring an energy flow of 250 Wm-2, the actual effective energy of the shell and the whole interior of the black body cavity would be 500 Wm-2.
What most people seem to forget is that a black body cavity is a reference not a reality. If you have a perfect radiant surface aka shell in equilibrium, the same amount of energy exiting will be entering, so the internal energy would have to be twice that of the exiting energy. A negligibly thin surface though would have no mass to speak of, so it would not be able to maintain energy as in forcing the black body to be in equilibrium. This perfect shell would just be a feature of a system in equilibrium not the cause of it.
Now for the Greenhouse effect simple explanation, that shell receives about 240 Wm-2 and radiates 240 Wm-2 at some point in the atmosphere. That point is "assumed" to be at the surface and negligibly thin by default since an ideal black body is used for reference. That requires the interior of that shell to be at 2X 240 Wm-2 or 480 Wm-2 if the system is in equilibrium like a true black body should be.
In the lab you have the luxury of forcing the body into equilibrium, adjusting the slit dimensions and thickness of the shell to measure what you want. You don't have that in the real world with a planet as a source so you can either stick doggedly to "your" concept or play with a number of references. For example, from the top of the atmosphere, the interior "should have" twice the energy and from the "surface" the shell or TOA "should have" half the energy.
From TOA the surface should be 480 Wm-2 and from a surface at 390 Wm-2 the shell, TOA, should be 195 Wm-2. The "surface difference is 90 Wm-2 and the Shell difference is 45 Wm-2 meaning that 45 Wm-2 is the black body reference effect, provided the "average" surface temperature is meaningful. Since the real surface has other forms of energy, latent, sensible, kinetic and potential that are internally never in "equilibrium" but instead in a sort of pseudo steady state, it is the more complex of the choices between the shell reference and the interior reference. 45 to 90 Wm-2 is a pretty big range for a reference used to determine a 4 to 8 Wm-2 impact. It isn't a complete obstacle, you can work some wicked statistical magic, but logically the "surface" in this case "shell" with the least amount of uncertainty would be less of an obstacle. Except for the little issue of that shell not being something that actually exists. The shell is an average of a turbulent fluid dynamics inside through and around a theoretical "TOA". You can estimate the energy of the "shell/TOA" but cannot actual define some surface as THE shell.
When you have what I call a fuzzy reference and you then try to set definitive values of the energy flows to force some example of "equilibrium" you open the door to Sky Dragons who are just pointing out defects in your choice of reference. Everyone should know there is a 45 to 90 Wm-2 "fuzzy" range and focus on the concept instead of trying to make believe it is a physical fact.
The infamous Earth Energy Budgets of Keihl and Trenberth illustrated that there is about 390 Wm-2 surface radiant energy, 17 Wm-2 convective or sensible energy and about 88 Wm-2 latent energy and assumed everything else is negligible. That produces a total surface energy of about 495 Wm-2 or about 15 Wm-2 more that what would be expected from a "shell" reference.
The climate scientists blow off the Sky Dragon efforts instead of explaining the uncertainties involved. The climate scientists also down play lower troposphere temperatures which are effectively measuring a different shell which would have a different energy, while they use a hybrid "surface" of mixed temperature measurements averaged of so large a range that the meaning of the average is in dispute.
Average incoming radiant energy is also dependent on which "surface" is being used as a reference. Since the "surfaces" are fluid and in motion you really need an average for each layer of fluid or a range instead of a fixed number. TSI/4 (ideal) versus TSI/pi()(oceans) which provides a range of 93 Wm-2 remarkably similar to the 90 Wm-2 difference you would expect by comparing "surface" and "shell" references.
None of this by itself will get you any greater certainty, but comparing the two does provide a useful range while eliminating most of the Sky Dragon arguments that are based on an overly simplistic understanding of the problem. S. Manabe indicated that the range of the GHE could be more than 60 C degrees depending on your reference, but that sound science seems to have been lost in the political battle to sell concepts instead of test them.
Because of the simplistic explanation there are plenty of non-issue issues. Evaporation for example doesn't "cool" the surface, it transfers energy to another surface which makes it easier for the energy to be moved over the surface particularly to higher latitude land area decreasing heat loss on a larger scale. Since this is latent or hidden energy in one location that becomes sensible and radiant in another, once again you have that pesky ~90 Wm-2 that is difficult to classify depending on your choice of "surface". The latent flux is just plain difficult to accurately estimate and once it becomes sensible energy the temperature change it produces can vary quite dramatically. If you use a term like "irreducible imprecision" which is actually pretty accurate, you see the eyes of your audience glaze over.
However, energy transferred from the surface or absorbed in the atmosphere increases the effective temperature of the atmosphere improving the insulation value of the atmosphere. Until you reach a radiant energy only portion of the atmosphere the plain old basic rules of thermodynamics are alive and well. Decrease the temperature differential and you decrease heat flow. This is actually tried in a number of passive and active or dynamic insulation schemes for homes but for the most part cost more than they save. With energy haters in charge though, that may change.
Some statistical magic applied to the TOA and the ocean heat uptake data does allow you to reduce uncertainty to a point, but since the TOA flux varies by 10 Wm-2 or more, the best estimate is still coarse enough to drive perfectionists batty.
Applying similar statistical magic to the "surface" temperature anomaly creates a false sense of certainty, because the relevance of the global mean surface temperature anomaly given the irreducible uncertainty just fuels the Sky Dragon fires.
In general, the great climate change debate in terms of actual physics is an exercise in futility since the reality is much more complex than the ideology. Pity, since the actual puzzle is interesting. Unfortunately, all this is pretty basic stuff which the climate science Gurus are beyond discussing. The neat thing about "good" physics though is there are always more than one way to skin a cat. Climate science currently requires just one series of assumptions which isn't necessarily "good" physics.
end of rant