When the wood rots, it also releases the carbon. Decomposition is essentially slow burning - so there is no difference in carbon between rotting or burning. The point about black soot is well taken, and there may be the counterpoint that particle air pollution may shade out some sun in the (very) short term?
Look at the Mauna Loa chart to see the rise of carbon dioxide when the trees in the northern hemisphere dropping their leaves every fall, and then going back down (less than it rose) when the leaves grow starting in the spring.
Why does the northern hemisphere matter more than the south? Because most of the land masses are in the northern hemisphere.
Making
biochar will sequester the carbon in the ground. So maybe burning wood, and then choking off the oxygen and creating unburned charcoal with part of the wood is a way to go? You would avoid burning oil or gas, and you could be sequestering some of the carbon as you go.