Carbon dioxide (CO₂) concentrations have increased in the atmosphere as a direct result of human activity and are at their highest level over the last 2-3 million years, with profound impacts on the Earth system. However, the magnitude and future dynamics of land and ocean carbon sinks are not well understood; therefore, the amount of anthropogenic emissions that remains in the atmosphere (the airborne fraction) is poorly constrained. As such, this work aims to quantify the sources and controls of atmospheric CO₂, the ultimate fate of anthropogenic CO₂, and the trend and robustness of the airborne fraction. We use Hector v3.0, a reduced form process-based coupled climate and carbon cycle model, with the novel ability to explicitly track carbon as it flows through the Earth system. This provides an unambiguous computation of metrics such as the airborne fraction irrespective of model feedbacks. We use a priori probability distribution functions for key model parameters in a Monte Carlo analysis of 10,000 model runs from 1750 to 2300. Results are filtered for physical realism against historical observations and CMIP6 projection data, and we calculate variance decomposition and the relative importance of parameters controlling how much CO₂ ends up in the atmosphere. We find that anthropogenic emissions are the dominant source of near- and long-term atmospheric CO₂, composing roughly 45% of the atmosphere. This is consistent with observational studies of the airborne fraction. However, when looking at the destination of anthropogenic emissions, only a quarter ends up in the atmosphere; most emissions are taken up by the soil. We also find statistically significant evidence for a negative trend in the airborne fraction from 1960-2020, implying that current-day land and ocean sinks are more than keeping up with anthropogenic emissions. This study evaluates the likelihood of airborne fraction trends and provides insights into the dynamics and final destination of anthropogenic CO₂ in the Earth system.