> CO2 from ancient sources doesn't have some special provenance, this is magical thinking.
Chemically, it is the same. The difference between CO2 bound in coal or oil to the CO2 bound in a tree is that the former would very probably remain bound for millions of years while the carbon bound in a tree will, with an overwhelming probability, will be released back in the atmosphere when the tree dies and its organic matter decomposes. And this will also happen if you bury the tree in the ground. It might not happen if you make charcoal from the tree and place the coal thousands of meters under the ground, where no oxygen is present. But this process is itself so energy-intensive, and therefore carbon-intensive, that it will not reach a net reduction of carbon.
Yes, in the case of a tree, it is theoretically possible that a vanishingly small fraction of the carbon it contains gets fossilized. But, there are at least two major problems.
Fossilization is very, very, very rare, and it is very, very, very slow. Told plainly, it normally does not happen.
The first is that you would have to plant so many trees that you would retain, as an extremely small fraction of the biomass of the tree, the amount of carbon released by your flight in geological deposits. This would take millions of trees and also millions of years, and is clearly infeasible. Nobody is even promising that, and suggesting that by planting a tree you could bury carbon permamently is just self-deception by people who buy such schemes.
And by that account, carbon-offsetting is just a scam. You pay money to extract the carbon you released by taking the plane from the atmosphere, but you can't permanently extract the carbon you released, at least not at a price that anyone is actually going to pay. It is a scam.
> Coal is just carbon that is in the ground. Grow a tree, convert most of it back to carbon, and bury it.
There is another factor which I find at times even more shocking. You suggested trees as a means to bind carbon. In fact, the fossil fuel from your plane flight is from oil and this is in large parts formed from marine sediments, also formed over geological times. What can be formed from trees is coal. Now, much of the large coal deposites we used today were formed in the carboniferous geologicial period, which was 358 million years ago:
https://en.wikipedia.org/wiki/Carboniferous
Now, the thing is that the geological conditions in which these coal deposits were formed do not exist any more:
https://en.wikipedia.org/wiki/Carboniferous#Rocks_and_coal
Basically, trees use lignin. At this time, lignin was something new on our Earth and there were no fungi which could decompose lignin, so the biomass of the trees could not decompose. Only later, lignin-eating fungi evolved.
That means that these coal deposits can never be re-created, even over a time of millions of years.