Sounds like a fun back-and-forth you have had there with your friend! By the way, I thought I would mention that Arthur Smith wrote up his arguments more formally and submitted the resulting manuscript [PDF file] to the same preprint archive as this paper itself is on. (I don’t know if he is going to try to get it published anywhere…It may be hard to submit to a refereed journal simply because it is responding to a paper that is extremely unlikely to ever see the light of day in any reputable peer-reviewed journal…and, as Arthur notes, the basic concepts are not new. It is just necessary to re-explain them after Gerlich and Tscheuschner [G&T] have confused them so.)
I agree with your sweater analogy in response to their Second Law of Thermodynamics argument in Section 3.9 of their paper although I have seen a defender of G&T object to such an analogy because he argues that the sweater is more just blocking convective heat transfer (just like a real greenhouse does) rather than radiatively absorbing and re-radiating it.
Leaving aside the issue of whether this matters, however, I think it is possible to come up with a system involving only radiative heat transfer that illustrates the same sort of effect and is so simple it could be given as a homework problem in an introductory physics course when they got to radiative heat transfer. (In fact, you could probably find such a problem in an introductory textbook.) Namely, you take three parallel infinite sheets in otherwise empty space that thermally-communicate only by radiation. They are all perfect blackbodies and one is held at a constant temperature. You could call the sheets A, B, and C but it will make the analogy clearer if we call them “Sun”, “Earth surface”, and “Top of atmosphere”. Of course, the sun will be the one we hold at a constant temperature, call it T_S.
First, consider the case when you only have two sheets. Then, it is a simple problem in radiative physics to show that the “earth surface” sheet in the steady-state will have a temperature T_E = 0.8409*T_S where the numerical constant is the 4th root of 1/2.
Now, go to three sheets (in the order “sun”, “earth surface”, “top of atmosphere”). Now, it is only slightly harder to show that the temperature of the “earth surface” sheet is T_E = 0.9036T_S and of the “top of atmosphere” sheet is T_A = 0.7598T_S where the two numerical constants here are, respectively, the 4th root of 2/3 and the 4th root of 1/3.
So, this is a very elementary radiation physics problem that seems to exhibit the exact same feature that leads G&T to claim there is a violation of the 2nd law: namely, the addition of the “top of atmosphere” sheet has caused the temperature of the “earth surface” sheet to go up from what it was in its absence and yet the “top of atmosphere” sheet is colder than the “earth surface” sheet so, by the 2nd law, heat should not be flowing from the “top of atmosphere” sheet to the “earth surface” sheet.
Of course, really we know that the net flow of heat is from the “earth surface” sheet to the “top of atmosphere” sheet…and thus there is no violation of the 2nd law. The “earth surface” sheet is only warmer now because the “top of atmosphere” sheet acting as an intermediary between the “earth surface” sheet and the vastness of space is slowing the net flow of the heat from the “earth surface” sheet out into space!