> I think CO2 would go down naturally after some time.

How? Where does the carbon go?

Eaten by plants, turned into coal. Atmospheric CO2 will indeed go down, after… some time.

My understanding is that weathering of rocks is bigger but slower than organic carbon cycle. It works on order of thousands of years, and is the main long-term balance.

I thought all coal was created in the carboniferous because no organisms had yet developed to break down that material?