Dead trees don’t evaporate (except for some of their water which isn’t too relevant); the decomposing (micro)organisms that eat them release CO₂ - about as much as burning the trees would. Yes, an old growth forest will be storing more CO₂ per unit of area than a newly created one but that does not increase very fast and after a century, the forest will have reached most of its carbon storage capacity it will be able to store for the next millenium - at this point, basically only oxygen-free peatbogs where dead biomass does not decompose (as the decomposers cannot breathe) will provide extra carbon storage. This will eventually turn into coal (and sunken sealife into oil), storing its accumulated energy as hydrocarbons that won’t be touched unless some pesky intelligent species start an industrial revolution.

Carbon capture is expensive: you need to isolate CO₂ from the atmosphere, which takes energy, and turn it back into solid or liquid chemicals, which by laws of thermodynamics takes more energy than burning that amount of corresponding hydrocarbons. Then they need to be stored somewhere where nobody will find and burn them to enjoy cheap energy from them like we’ve been doing for the past 200 years. So we’ve had cheap energy at the expense of the environment, and we need to expend at least as much energy to get all that nasty carbon back out of the atmosphere. Thunderf00t made a very informative video about all this.