At least this science survey concludes that old forests do accumulate carbon. https://www.nature.com/articles/nature07276

Also a lot of the forest mass that gets cut is used for "green energy" so it's worse than letting it stay in the forest. Basically in a periodically burnt (bioenergy) forest the carbon is kept in the atmosphere most of the time while in an old growth forest it's in the trunks all the time.