Just to be clear, how are you getting frozen nitrogen in Antartica? The delta T between the ground and space may be about 273 degrees, but there is a large atmosphere in between that has a huge thermal resistance, because of the low density. The air near the surface of the Earth will do a much better job of convecting heat laterally, and the ground will store thermal energy from the previous day. These are the same reasons why the dark side of the planet in RET stays warm. In FET the sun moves around and partially illuminates Antartica once in a while just like RET. It is pretty difficult to say that heat transfer in concept alone defeats any possible FET.
The funny thing is, to be serious FET would have to propose a model of how the Sun might move, and that model could be tested. For now there is no serious FET model so there is nothing to apply math to! Until then there is no FET to really disprove, more like FEC "flat earth concept". Because of this it is so easy for them to keep changing the position and size of everything. There is no real model, only bits and peices that don't fit together.
Your point is very important. If we compare, say, the Earth around the Equator and the Earth around the South Pole, both lose energy at a rate that is proportional to the current temperature and thermal conductivity and inversely proportional to the greenhouse effect. Whatever these parameters are, the thermal conductivity and the greenhouse effect are similar everywhere on Earth since all parts of the atmosphere have about the same gases in the same proportions.
But because the loss of heat is proportional to the temperature and the temperature is proportional to the irradiated energy from the Sun, the average temperature at the Equator and at the South Pole are both proportional to the irradiated energy from the Sun. Both could be a lot higher, as we see in Venus, or a lot lower, as we see in Mars. But in both cases it is proportional to the irradiated energy from the Sun.
Your concern would be justified if somehow the greenhouse effect or the thermal conductivity of the atmosphere were substantially different in different parts of the planet, but this is not the case. Also, the average temperatures of each part of Earth has had millions of years to stabilize.
And yes, instead of a model from which predictions can be made, FE'rs revel in their ability to change their "models" as fast as anyone can analyze them, making predictions hard and lame excuses easy. But thermodynamics give no breaks to the sloppy scientist.