Modified blackbody model of earth. What is wrong with it?
The premise that the Stefan-Boltzmann equation is used for greenhouse gases is wrong. The Stefan-Boltzmann equation is used for irradiance output, not energy input. The ground warms the atmosphere, and the atmosphere releases energy into space.

I suggest looking at the one-dimensional energy balance model. See http://kurs.uib.no/acdc/filer/219.BYQINh.pdf