Breathing is the intake of air in to the lungs, where gaseous exchage occurs via the alveoli. [...] Believe what you want, but [sucking in more air by consciously using the muscles of your] belly [when] breathing does not give you a significantly higher volume of oxygenated blood.
But wouldn't sucking in more air (through conscious use of the belly or otherwise) dilatate the alveoli further, thus increasing the exchange surface and/or facilitating the said exchange by stretching the alveoli walls thinner?
What you say makes sense and you definitely know your stuff, but still I find it hard to believe that deeper breathing (breathing rates being equal) doesn't lead to a higher supply of oxygen...
A hidden child playing hide-and-seek will see his breathing rate increase as the seeker passes him by (strictly 'mental' stress condition, ready for a 'fight-or-flight' situation but not quite there yet), while someone whose body actually has a clear, physiological, immediate need for more oxygen, like an olympic sprinter for instance (actual physical stress conditions) will not only see his breathing rate but also his breathing 'depth' increase. The first one's breathing is fast but shallow and silent, while the second's is fast, powerful and noisy.
What would be the point of that, physiologically speaking, if increased breathing-depth doesn't provide the body with more oxygen?