Great thing about wood is you can build almost-permanent structures with it?
And the remaining biomass (leaves, bark, etc.) is fine to compost and re-enter the carbon cycle.
Also it's clearly not an open cycle, not in temperate climates. The accumulation of carbon in top soils etc. is one of the things that kept CO2 in balance for millions of years prior to us pulling it out of the ground. Peat bogs being another key one.
Yes, it can't keep up with us. But it's not an open cycle. (It is in the tropics, though)