Before we blindly start building everything in timber we need to ensure it is done sustainably. Logging accounts for ~26% of deforestation annually, considering 15% of global emissions are from deforestation this means that logging is responsible for ~4% of emissions. While still better than steel or concrete, deforestation is a double whammy that decreases the carbon sink (while also driving biodiversity collapse which is a huge risk for humanity).
Overall we need fewer buildings, make the ones we already have last longer, and eventually short rotation agricultural crops waste could be used to sequester carbon in buildings.