Why? We give oxygen to injured people because their lungs are impaired. But you can’t force more oxygen into the blood than it can carry, it just doesn’t work like that.
Uh, incorrect. Resting o2 sat is 99% in most young healthy people - like aviators. In older people, or the unhealthy, you may see it start to drop due to diffusion or ventilation defects - but barring serious lung disease, neuromuscular defects, severe, severe obesity, etc. that still gives you a resting sat in the mid 90s.
There is also oxygen dissolved directly in blood rather than saturating hemoglobin, but that constitutes a tiny fraction of o2 in comparison.
-Oh, breathing hard when exercising is a thing, noone claimed otherwise. Your body requires more oxygen. You breathe harder. Heart rate increases to bring that oxygen to your muscles. That still does not oversaturate your blood with oxygen, though - only maintains saturation as consumption increases.