In their natural form, you are correct. But if you cut down those trees and put them in Wood Vaults, specially engineered enclosures to ensure anaerobic environments, thus preventing wood decay -- then you would properly sequester the carbon. That is literally what happened in the carboniferous period that caused all the trees to turn into coal.
There is recent research on this that indicates we could achieve a 10 gigaton annual sequestration rate with only a 5% impact on the total terrestrial tree production and at a cost of $30/ton.[1]
"The quantity of this wood utilization can be controlled carefully to maintain a desired amount of CO2 in the atmosphere to keep the Earth’s climate from diving into the next ice age, acting as a climate thermostat."
Can AI improve on evolution? Tree DNA and cells must be super complicated, but they're evolved things so they must be inefficient. Can we figure out how to do the same things they do, just faster? Bamboo can grow inches per day, can we make a petri dish that grows centimeters per day of carbon capture?
I like the idea of using novel biological methods to capture carbon. Maybe we will end doing something like that, and I know you're just spitballing here, but the economics of scaling microbiology tools is very wasteful in terms of environment.
The inputs for petri plate growth have a demanding level of refinement. Pure sugar, yeast extracts, salts, gelling agent, all dissolved in D.I. H2O. Sterility requires autoclaves which are energetic monstrosities. Without these conditions the culture will become contaminated and overgrown with stuff you don't want.
Innovation could solves these problems, but it's going to be hard.
Those are not a viable solution as they don’t give politicians any power. Imagine what good it would bring to society if we were able to have a grow / harvest cycle that would give us an abundance of inexpensive lumber to build housing.