We have a number of clean burning wood stove requirements. I don't know the overall impact, but it does reduce the visible smoke a lot, and I believe also reduces any harmful particulate matter.
The carbon cycle is very different from fossil fuels.
For the fossil fuels, the carbon has been sequestered for millions of years. Put it in the air, and does anybody want to wait a few million years to get it buried again?
For the tree, perhaps sequestered for a few dozen years, or a century or two. I'm not cutting any living trees at the moment. Most of my cutting is something that has fallen to the ground in one manner or another. If left on the ground, that wood will rot, and over a few years, or perhaps even decades all the carbon will be released by the bacteria, fungi, and various borers that feed on it.
But with the carbon cycle, if I cut something down, then it regrows, then over a period of time that carbon is sequestered again. Or, perhaps if I simply thin a few trees, then the growth of the neighboring trees will make up for the lost trees.