The Bohr effect describes how binding affinity for O2 to hemoglobin decreases with reduction in pH. This is most beneficial for understanding oxygen unloading in tissue where pCO2 rises and increases the efficiency of oxygen delivery.
What you are talking about is diffusion vs. perfusion limitation. And yes at 1atm both CO2 and O2 are perfusion limited. Meaning the gas exchange across the alveoli occurs more quickly than the blood flows through the length of the capillary associated with the alveoli. if you experience diffusion limitation at 1 atm odds are good you are not diving because if you have a condition that leads to this a physician will not clear you for diving. Given O2 is perfusion limited at a partial pressure of 0.21 atm it is then quite clear that at depth there will be no issue saturating the blood with oxygen. In fact, as we often discuss it is the inverse we concern ourselves with in oxygen management (ie hyperoxia).
However, CO2 is being produced at a constant rate (relatively) and must still be eliminated. But now we’ve changed the dynamics of this substantially and this is seemingly largely related to the densities of the gas. CO2 is elimination is dependent on minute ventilation. That is tidal volume x respiratory rate. At depth with increased gas density the air does not move as freely through the lungs and gas exchange is impaired. The physiology of this does not appear to be fully understood from my reading of the literature. However, suffice it to say we do not eliminate CO2 as efficiently at depth. Further, our basic compensatory mechanism of increased minute ventilation through increased respiratory rate does not appear to work well given the dynamics of gas exchange at depth. The elimination of CO2 as optimized by oxygen displacement at the level of the alveolus is explained by the Haldane effect. But this mechanism seems to be intact at depth.