I did not see in Hills criteria any consideration of alternatives. CO2 is not the main greenhouse gas. Variations in orbital inclination and cosmic rays, as well as solar variations, have been proposed as causing the climate change but here we only discuss CO2.
I also think the earth’s long-term history should be considered as well. Keep in mind that the earth started out with very high levels of CO2 and little if any free oxygen. Plants and photosynthesis changed the atmospheric composition to the very low levels of CO2, and high levels of O2. It is quite likely that all the coal in the ground was once CO2 in the atmosphere and only became coal over the millions if not billions of years the plants have been photosynthesizing CO2.
I suspect that with lower CO2, plants are starving, higher levels of CO2 mean plants need less water to survive. I would like to see some indications that in the long term history the planet had higher temps to go along with the higher CO2. As far as I know, in geologic times of about 70 million years ago, the CO2 was much much higher yet plants and other life thrived. Burt Rutan’s analysis claims that 6–8 times as much CO2 was present but temperatures appeared to be fairly consistent with today's.
And one other point regarding the calculation of the earth’s temp without considering the atmosphere. I have done that calculation myself. But no one has included to my knowledge the internal heat of the planet. I would have to go find the relevant heat transfer texts to figure it out but the internal temperature of the eath’s molten iron core has an effect on the surface temperature which is where it radiates into cold space. That has not been adequately addressed to my knowledge.