No, this vision is (mostly) incorrect. In more detail :
Any molecule having a non zero electrical dipolar momentum will interact with the electromagnetic radiation. In the QM vocabulary it will absorb and emit photons.
The CO2 molecule has 3 vibration modes and 2 of them produce an electrical dipole. The vibration energies of these modes are quantized and have energy levels En. For the first excited mode E1-E0 = h.f provides the 2 photons frequencies that will be absorbed/emitted at strongest. Their wavelengths happen to be 4µ and 15 µ what puts them squarely in the IR spectrum where Earth radiates.
So the first part of your vision was correct. CO2 but mostly H20 strongly absorb and emit photons in infrared. Because of Kirchhoff's law, this activity is a zero sum game, e.g the CO2 molecules absorb exactly as much as they emit. There is therefore no net heating of the atmosphere by IR.
There is no net energy transfer by "thermalisation". Indeed a vibrationnally excited CO2 molecule may collide with an N2 molecule, decay from E1 to E0 and transfer E1-E0 to the kinetic energy of the N2 molecule. However the inverse process exists too - an N2 molecule transfers E1-E0 to an unexcited CO2 molecule making its vibrational energy going from E0 to E1. In steady state both rates are obviously equal and there is no "heating up". Also common sense tells us that if there were a net energy transfer, the N2 molecules' temperature would diverge and reach an ultrarelativistic plasma rather fast.
It follows that the collisional processes are also a zero sum game
The right vision is then that in a mixture of 1 GHG gaz (non zero dipole like H20) and 1 non GHG gaz (zero dipole like N2), the GHG gaz will absorb and emit in the infrared spectrum and it will absorb exactly as much as it emits. The non GHG gaz lets everything pass through. Beside the radiation process which involves only the GHG gaz, there are collisions that involve both the GHG and the non GHG gaz. The role of this process is to make sure that the radiatively active GHG gaz and the non radiatively active gaz stay both at the same temperature. In steady state there is then no net energy transfer by collision between both species.
The fact that a GHG - non GHG mixture will be warmer than the case with non GHG only is an effect of density of radiation. Indeed in a non GHG atmosphere the radiation energy density is constant from the bottom to the top because the radiation goes through with a constant rate. As the matter doesn't interact with radiation, it just adds its own kinetic energy to the overall energy density.
In the case of GHG - non GHG mixture, the absorption/emission processes have for effect to decrease the photons' mean free path and thus to increase the radiation energy density as compared to the non GHG case. The GHG matter interacts with radiation in this case and its equilibrium temperature will be higher than in the non GHG case because of the higher energy density. The collisions will then make sure that the non GHG gaz will be at the same temperature as the GHG gaz. The overall result is that the whole GHG atmosphere will be at a higher temperature than the non GHG atmosphere.
In the above I was describing the radiative and collisional processes only. In the real atmosphere add convection, phase change (evaporation, condensation etc), albedo changes and conduction. There is no reason to believe that these energy transfer processes are not impacted by radiation. Therefore making the jump from a GHG-non GHG mixture to a real atmosphere involves an implicit "all other things being equal" hypothesis which allows a qualitative conclusion but would be wrong for a quantitative conclusion.