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I'm not an expert on this subject so let me see if I can determine exactly what you mean.
When you say the oxygen is the same at higher elevations, I assume you mean it remains 21% of the existing pressure (or amount-of-air-per-given-volume), but the partial-pressure of oxy is lower because the overall pressure of the atmosphere is lower... right?
Not 21% of the existing pressure, 21% of the content. The air we breathe is 21% oxygen. That percentage does not change at 24,000ft, but the pressure does. So whether you're at sea level or 24,000ft, 21% of the atmosphere is oxygen, but the pressure gradient is vastly different.
Medical science does not yet know why this pressure makes a difference for breathing, given that the oxygen content is the same.