`it's not the rotten timber, it's the timber that would rot if it were left there, and apparently there is enough of it being gathered in some places to be a concern about the ecology.... but I doubt that many people who use a wood burner as a principal source of heating are fuelling it with rotten wood gathered from a forest floor.
Where the timber comes from is of concern. If it's "recently grown" farmed wood then that's OK (but as you hint at, there's still the net carbon released during it's planting, harvesting, preparation, and transport. But, I get the impression that a lot of wood coming from northern US and Canada isn't really farmed in the way our conifer plantations are - a very large part of it is what's been growing (slowly) for a long long time, and the replacements will be many many decades in growing back to the size of what's been harvested.
Still, as you say, it's a lot better than coal/oil that's been thousands or millions of years in the making.