I will continue to make the argument about all "greenhouse" gasses, as that is the largest scope of applicability; it necessarily follows that if any "greenhouse" gas will cool the surface, so will CO₂.
I will now define "greenhouse gas" and stop putting it in quotes so that it can be understood that even though I'm using the word it isn't much like a greenhouse at all.
Greenhouse gas - A gas with significant spectral absorbance in the range of the blackbody radiation of a celestial body while having insignificant spectral absorbance near the peak of the local star(s) radiation.
Earth's blackbody radiation was shown in the graph I linked to many times 210k-310k (Average is 15C 285K). NOTE! This is the surface of the Earth as in the land and water, not the top of the atmosphere.
We'll use blackbody radiation of 285k for our presume surface radiation.
The sun will have a blackbody temperature of 5777K.
The tools for equations and spreadsheets do not easily translate to hyper-anonymous web activity. Therefore I will make use of some calculators so the reader can follow along:
The surface area of Earth is 509,600,000 km² = 5.096×10^14 m²
The surface area of Sol is 6.09e12 km² = 6.09×10^18 m²
Earth (the ground) radiates 1.906403805179055e17 W. 91.21% of that energy would be between 6000 and 40,000 nm with the peak being at 10.16 microns.
Sol radiates 0.384620829026384e27 W with a peak being around 501.6 nm.
Let's sanity check by calculating the irradiance at Earth's orbit. To do this we consider the total power of the sun at the average orbital distance of Earth (r = 1AU = 1.495e11 m).
The ratio of the power divided by the surface area of the sphere of radius 1AU:
This is close enough to the wikipedia "1361 W/m^2" to assure us no order of magnitude mistakes are present so far.
Let's now use that calculator and these same calculations to determine how much of that irradiance is in the band of the blackbody radiation of Earth's surface.
The solar power radiated between 6000 and 40,000 nm is 1.2999051628954013e24 W. This is over a thousand times less than the total, so right away you can tell it's not going to be much of the irradiance.
That's only 0.34%. That is how much power would be blocked at the top of the atmosphere if the atmosphere was "perfectly greenhousey".
But what about the other side of this hypothetical wall?
Radiation from the radiating surface in question (earth-sea) would indeed be scattered by greenhouse gasses, and the effect of having less than saturation might well be noticeable cooling effect.
However, beyond saturation the effect would extremely rapidly reach: nothing.
Is Earth at saturation? Yes. How do I know? The graph. How far above saturation are we? A considerable amount from my memory of previous research, hundreds of times for water, more than twice for carbon dioxide.
That means when it says "100%" it's more than 100%, an average photon will not just hit a resonant electron shell once, but multiple times on a path outward (or inward).
Every time it does this is a chance to be converted into kinetic energy (when a collision between particles occurs while an electron is still excited).
In other words thermal photons don't make it from the surface to space (almost always). They serve only to bonce around and dissipate into kinetic energy (from whence they came).
They have never done anything different because this planet has had saturation carbon dioxide and water for every time period with life in it.
Mars is in saturation (despite its thin atmosphere), Venus is so far above saturation that you can bet a 6-40 micron camera sees only total darkness a few hundred meters below the clouds.
So if photons from the surface aren't cooling the surface and Earth must cool by radiation in this 6-40 micron band then how does it cool? I've said this already, but let me put it at the end of a post so as to make sure it isn't lost:
The energy is transported by convection (and conduction and radiation) to the upper atmosphere. There, where the radiation does not need to go through a fog of greenhouse gasses (water most of all) it radiates out into space.
Therefore the very slight effect of increasing greenhouse gasses is to slightly increase the temperature of the upper atmosphere (moving the interception layer higher) while slightly decreasing the temperature of the lower atmosphere (and surface).
Now, if you ask for the exact dynamics of convection, clouds, temperature, and water vapor I will tell you I do not have that and that nobody has that. That is the poster child for chaotic systems, it's also known as the weather and nobody anywhere can predict it. Suffice to say that the way it operates has no dependence on the radiation ping pong of trace gasses.
I do not claim to have all the factors or make predictions about the whole, I don't need to; my claim was about the specific effect of one variable changing.
Speaking of clouds, a slight decrease in clouds could very easily account for the increasing temperature and be nearly impossible to quantify. How much less useful (politically) is "the clouds might be doing something, we don't know why".