The carbon cycle is complex and simple half life numbers are always wrong. A lot of the carbon we emit is removed from the atmosphere relatively quickly, for ex

The carbon cycle is complex and simple half life numbers are always wrong. A lot of the carbon we emit is removed from the atmosphere relatively quickly, for example into the biosphere or into the oceans. But the only way to "permanently" get rid of it is by sequestering it into the lithosphere. These processes take geological time. Until they are complete the equilibrium concentration in the atmosphere is larger than before we burned that piece of coal. So it is relatively accurate that some of the carbon we emit stays in the atmosphere "forever".