CONCLUSION:
The surface conducts "heat" to the air, not the other way around...unequivocally. Ergo, by extension, atmospheric gases--including CO2--COOL the surface via conduction.
AGW Scorecard:
Conduction FAIL
Convection ?to be determined
Radiation
First by definition, energy transfer between the surface of a soild and any gas like environment is convection. Conduction is general considered via a mechanical link between solid or liquid bodies. Trivial but important.
I think you're poking down the wrong rabbit hole.
The issue isn't about where the energy to heat the planet is coming from, but rather what attribute is responsible for attenuation of re-radiation of incident energy. Mind you, that is only an issue if you believe, as some do, that the attenuation has increased as a function of the habits of man.
The claims re CO2 being the major modulation factor contributing to attenuation is the issue. So the discussion should be, what is it modulating and precisely how is it accomplishing this? But that only matters if there's a reliable reference to base the claim that long term thermal changes are actually occurring to begin with.
The problem I have with all this is that backward looking simulation can be designed to confirm anything one desires. Once predictions with accuracy and repeatability looking forward are demonstrated, then and only then, can one consider the science mature.
We aren't there yet.
.