The question is not whether it is a good idea for greenhouse gases to be regulated. The question is if the Clean Air Act mandates it. IANAL, but from what is quoted, I would have a hard time classifying CO2 (which already makes up 0.04% of air) which you exhale every time you breath as an air pollutant. There is no evidence that breathing air with CO2 (at the levels that current industrialization produces) produces harmful effects directly. In fact, if you hold your breath and exhale, the air you exhale likely has a higher local CO2 percentage than the increased levels caused by industrialization. The effects are mediated by its greenhouse effect and global warming. This is analogous to the CFC and the ozone depletion. The CFC's were pretty innocuous at ground level, but their release cause ozone depletion. This was regulated not through the Clean Air Act, but by the Senate ratified Montreal Treaty (https://www.nytimes.com/1988/03/15/science/treaty-on-ozone-i...).
The issue right now is that there is not the political will in the United States to address greenhouse gases, therefore people are trying to accomplish this through a very broad reading of the Clean Air Act. This will not be helpful in the long term as it just further associates fighting climate change with government overreach, and further politicizes what should not be a partisan issue.