The adiabatic lapse rate is established by molecular collisions (diffusion) which must retain entropy. The Second Law is not violated just because it appears that heat is transferred downward in the atmosphere. It’s not. All that happens is that PE converts to KE, and entropy becomes uniform over the whole system, be it a small room or the whole atmosphere.
If, as is generally the case for perhaps over 99% of Earth’s surface, there is a small “step down” in temperature from the surface to the atmosphere at the interface, it is not possible for KE to be conducted (diffused) back from the cooler atmosphere to the warmer surface.
Yes, the base of the atmosphere has, over the life of the Earth, autonomously achieved an equilibrium temperature determined by mean Solar insolation and the effective adiabatic lapse rate which is proportional to the gravitational force.
However, where the surface is warmer its temperature is merely “supported” by that of the base of the atmosphere. It cannot fall significantly below that temperature in calm weather over a flat surface, but as the temperatures approach, the rate of conduction (diffusion) from the surface to the atmosphere will slow down – that’s standard physics. Rates of evaporative cooling do likewise.
The slowing of these non-radiative processes depends on temperature differences right at the interface. In contrast, the slowing of radiative cooling (less than a third of all surface cooling) depends on the temperature of the source of radiation (which may be much higher in the atmosphere) and the proportion of the total potential Planck curve for that temperature which is actually “filled in” by the spectral lines of radiation. This is why carbon dioxide has less effect per molecule than water vapour, and WV less than a solid surface which may be close to a true blackbody if it has a wide range of elements in its composition.
But, as I have said, any slowing of radiative cooling will be nullified by a compensating effect in non-radiative cooling. The pre-determined lapse rate is the main player in adjusting this compensation process.