Not just houses but any kind of building. There is an architectural movement called Mass Timber where the buildings are not just framed in wood but are virtually solid wood. It actually solves a lot of problems around building cavity thermodynamics, by eliminating the cavities. A large scale mass timber building sequesters a huge amount of carbon.
Carbon sequestration is a nice side benefit, but wood-into-durable-products can absorb only a small fraction of present anthropogenic CO2 emissions. We're emitting about 37 billion tonnes of CO2 per year at present (10 billion tonnes of carbon) and wood is about 50% carbon. To offset a quarter of human CO2 emissions, we'd need to turn about 5 billion tonnes of wood per year into long-lasting buildings and other manufactured objects. As of 2014, the world was producing about 0.8 billion tonnes of wood products that could be used in long lasting applications (sawnwood and wood paneling).
I see what you're saying, but that doesn't actually strike me as too crazy of a goal. 0.8 is 16% of 5, and 6x-ing an industry seems within reach, given sufficient incentives.
There is enough landmass in Northern Canada / Russia to do this on a mass scale. Could you coat the trees in something to greatly reduce their likelihood of burning? Therefore not having to use biochar or some other form of carbon storage.
You would also have to coat them in something to prevent decomposition. Assuming that worked, over time you would take a ton of nutrients out of the system and the primary forest would eventually become grassland.
Maybe if you were growing trees specifically for this purpose on degraded land it could work.
