Increasing the world's forest cover is worth doing for its own sake, on biodiversity grounds, but its at best a small part of the solution to global warming.
Not if they keep growing, which if there's plenty of CO2 to feed them, they will.
If carbon-free energy were money-free...
I'm still accepting funding in the from of rare stamps.
If you wanted to zoom out and squint and get a little biblical, this is a Garden of Eden situation. There was an atmosphere that could not support our lives. There was a perfect mix of things at the same time to change that atmosphere into the one in which humans flourished. It was something too powerful we'd never be able to replicate ourselves. All we had to do was not exploit buried hydrocarbons. But we've instead made a race to dump as much carbon in the atmosphere as possible, it would seem. And now the comfortable world we live in will doom us to live outside of paradise.
Unless you're doing deep carbon extraction, nature is pretty well carbon neutral. A forest of some given density, over time, will remain carbon neutral. If it gets thicker, the carbon captured is more, but it's also more prone to fires (which obviously then release that carbon).
Nature is circular. Any "waste" from one process is an input into another process. It's humans that think in linear "Resources into products into waste" ways.
But outside equilibrium, the forest canopy will expand until equilibrium is reached, or the forest canopy will shrink until equilibrium is reached. You will also see more hungry plant life supported in environments with more carbon again until starvation levels are reached.
It is like any other kind of food. We can think of food as sequestered in the living bodies of a population, with deaths matched by births, a constant amount is sequestered. But increase food and population goes up until starvation levels are reached and now more is sequestered. Decrease food and population falls so less is sequestered. It doesn't matter where the food comes from. Currently 20% of the earth's carbon is sequestered in plant biomass. This is why various carbon offset programs do include increasing forests as a legitimate offset, but the land has to be allocated to the forest in perpetuity. It's not like you can grow 10 trees, the point is to support a bigger forest where there are 10 more trees permanently.
Thus nature regulates carbon levels at those altitudes that trees can feed from. I have no knowledge about equilibrating mechanisms in the atmosphere as a whole, this discussion is for carbon accessible to plants.
No, you can get it to sequester carbon if you plant trees (or other plant matter), harvest it, convert it to charcoal, then spread that around. Apparently in that form (biochar) it stays sequestered for a few thousand years: https://en.wikipedia.org/wiki/Biochar
Left alone, nature's cycles are mostly carbon neutral.
Obviously violent combustion will put a great percentage in the air, but I don't see a strong reason to believe that a rotting tree trunk will completely put carbon in the air and none of it in the ground.
All the carbon that makes up most of the mass of plant is from the CO2 in the air. Decay and rot release whatever carbon was captured back into the air.
The same way the food you eat turns to CO2 you breathe out but due to fungi and bacteria.
Not all tactics will be equally effective. It's worth investing in the approach(es) that are most effective in proof-of-concepts trial runs guided by a first principles perspective. That's how to maximize change. GMO kelp and phythoplankton for oceanic BECCS seem like the leading candidates.
Doing "something" is futile if it's ineffective, not permanent, and not scalable.