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?
Thats not how it works. The best way to think of alveoli is like tiny clusters of blackberries - thats kind of what they look like, all lumpy. This allows them to have a greater surface area. The lungs, if completely stretched out, have the same surface area as a tennis court, and that is due to the space saving structure of the alveoli. The air passes around the alveoli, which have membranes so thin, that they absorb the properties in the air and these go straight to the blood stream.
lung capacity, lung power, and lung efficiency, are seperate qualities, with seperate measurable tests. Someone with lung damage from smoke can breath as deep as they like, it will not change the damage to the alveoli, nor the resultant limitation in lung efficiency. I have also seen many with very strong peak flow readings fail badly on Vo2 max testing.
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...
'deep' is a relative term - again, a top level free diver, or opera tenor still doesnt use much over 1/3rd of the lungs potential expansion ability. So sure, increased volume of respiration will process more oxygen- its why you get 'out of breath' from exertion, but oxygen debt recovery difference from slower consciously deeper breathing, and more rapid natural breathing is swings and roundabouts situation.
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),
But why? This is the crux of the argument. That breathing increase, the pulse increase, the adrenaline 'jolt' as you say are all pre-emptive physiological changes. They prepare for exertion, they are not stimulated by exertion. As such, in this prepared state, deep breathing is not 'paying back' an oxygen debt - the blood is already highly oxygenated, and no exertion has occurred. The deep breathing is a message from one element of the emergency system to all others, that the threat is not real, or is over - ever heard the phrase 'to breath a sigh of relief'? Thats why.
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.
Probably a bad example, as sprinters breath very shallow, as their exertion is anaerobic and brief. Though this is a good time to introduce the impracticalities of 'belly breathing' to situations of gross motor function. A sprinter relies on a stable core through which to ensure that the least amount of power dissipates through yielding body mechanics. To truly 'belly breath' the abdominals must relax beyond a point where they can effectively stabilise dynamic physical movement. This is why traditional meditation takes place in a static position.