No, I am not missing this part at all.
The problem with the way you put it is that it does not include the element of time. And this is massively important.
The CO2 emitted by burning a forest the size of a city in just a few days isn't recaptured at the same time scale. In fact, the time scale looks to be orders of magnitude greater.
Forget about humanity for a moment (which is easy to do if we go back 50K, 100K, 200K years. Yes, CO2 was emitted through many mechanisms and re-captured. One of those mechanisms was massive continental-scale fires. There was nothing whatsoever to oppose them. No modern fire-retardant carrying aircraft, etc.
The natural rate of change for atmospheric CO2 is in the order of 1 ppm per 1000 years. Source [0], graphs [1], rate of change [3].
This rate of change --at a planetary scale-- exposes both the realities of natural atmospheric CO2 accumulation and capture.
From there it is somewhat easy to calculate the time required to recapture any CO2 added to the atmosphere. Since we don't have any technology today that can beat the natural rate of change, the only reliable metric we have is 1 ppm per 1000 years. CO2 doesn't magically hang around forest fires until, 20, 30 or 40 years later, it is recaptured. It contributes to global warming while trees are growing, for a very long time.
Side note: Lab experiments are irrelevant. We need something that works at a planetary scale. Nobody has shown a solution that can pass a physics test working at that scale.
Here's a simple article essentially saying the same thing [3]
Quoting:
"The idea that forest biomass is carbon neutral is not wrong. Trees absorb carbon from the atmosphere as they grow. So if a tree is burned for fuel, the thinking goes, another can be planted to replace it. And then that replacement tree should eventually re-absorb the carbon.
But while burning wood immediately releases carbon, it takes decades or even a century for a replacement tree to absorb the same amount of carbon. In the meantime, all that atmospheric carbon will continue to drive climate change."
That is the problem. You are not including time as a variable in your mental model of how this works. That time scale is easily in the hundreds of years, particularly if you include what happens with soil, root system and natural vs. managed forests (as discussed in the article).
Time is, perhaps, the most important variable.
[0] https://cdiac.ess-dive.lbl.gov/trends/co2/ice_core_co2.html
[1] https://cdiac.ess-dive.lbl.gov/images/air_bubbles_historical...
[2] https://i.imgur.com/37AKa8L.png
[3] https://www.smithsonianmag.com/smart-news/epa-declares-burni...