Don’t feel guilty about harvesting trees. This is literally a carbon capture business. They are not cutting them down from the wilderness.
Even if you're not burning it the day after Christmas, it'll get chipped and composted. Some of the carbon is released in composting, some wood will remain, but with how people use mulch, I wouldn't expect it to capture carbon long enough to be interesting.
The alternative is to gasify it, and produce biochar (a.k.a terra preta), a form of carbon that is mixed with the soil to improve it's nutrient and water retention, friability and microbial activity. Biochar is theorized to be stable for 10k years in the soil.
I went to the tour of recycling plant, and as part of the tour they told us what happens with Seattle garbage: they excavate a pit, line it up with plastic, put garbage in, cover it with plastic, put soil on top. Without supply of oxygen it basically stays there almost forever. They cut a carrot with was in such pit for 10 years. It was black on the outside, but still orange inside!
Anyway, put trees in such pits, and you have carbon capture!