How Te and T mean change with an atmosphere compared to no atmosphere.

This is actually very good and effective reasoning with a salient point at the end.
Worth publishing?
” That is why the effective temperature for Moon is so much higher Te =271K, than the satellite measured Moon’s mean surface temperature Tsat = 220K.
It is because they have not considered the Moon’s specular reflection.
As a result the theoretical Te for Moon was overestimated by 271K – 220K = 51oC”

Both the earth and the moon receive the same amount of radiation per square meter. 1361 W/M2 fact.

Both [reflection and emissions added] send 1361 W/M2 back into space per meter of disc or a quarter of this averaged over a sphere or half of this per meter averaged over a hemisphere. fact

The albedo of the moon 0.11
The albedo of earth 0.306 fact.

Specular radiation does not change this. Fact

The TE of the moon Te 270.4 C and that of the earth is Te 254 C due to the difference in the amount of light reflected only. Fact.

The Tmean and the Te for the moon surface are both measured at the moon surface as with virtually no atmosphere They are at the same level which is therefore the Moons TOA
The T mean is lower because the moon surface respective to the sun rotates extremely slowly hence the average of the cold and hot side surfaces is always lower than Te. Fact

The Tmean and Te for the earth or any planet with an atmosphere are measured at different levels.
A square meter at the earth surface is much larger at the Te height at the TOA.
Hence the surface of the earth radiates more energy per square meter and is hotter than the TE at the TOA where it is measured.
Fact

This is why the Earth is 33 degrees warmer.
It has an atmosphere with GHG whose absorption and emission profiles exactly match that needed to make the temperature differential occur.
Exactly. Fact

That is why we have radiative science and are able to measure temperatures from satellites. Fact

This would happen whether the planet is rotating or not. Fact

A rotating planet helps push the Tmean up to the Te for an airless planet
A rotating air planet is warmer than an airless planet at the surface
The Te does not change for an airless or air planet

Note the following conundrum or corollary and proof 17/3/2022.

Theoretically we could try imagine an atmospheric planet with a Te less than the Tmean.
All planets with no atmosphere have a Te higher than the Tmean.
All planets with an atmosphere should have a Te lower than the Tmean
Where does the crossover point occur?
The ingenious answer to this must be as soon as one designates a planet to have an atmosphere.
The Te must rise above the surface of the air planet yet fall below the Tmean of the the surface whether it is rotating or not.
This means that the TOA will always be higher in a rotating air planet.