I am actually about to do a video along the lines of “the air goes in to the lungs” for the Allegro Status Stage. What we do with the ribs and musculature around the lungs can determine how much the lungs expand and therefore the quantity of air in the lungs.
If you relax your abdominal muscles and let the belly fall out (not pushed or forced out), the diaphragm drops allowing more air in to the lungs. For louder, longer notes, abdominal muscles become involved, raising the diaphragm increasing the flow.
It does not take very much air to play a brass instrument and your sensation may be misleading you. If you are doing the BCH breath slowly and softly and allowing the body to relax when you do it, you are getting plenty of air in to the lungs.
I encourage you to have a look at an anatomy picture and look exactly how high the lungs are in the body and really appreciate the sensations of “low breathing” of which I took about in M2MII. It is simply about getting an efficient breath easily and quickly.
The “urge and necessity to push” is what I am trying to reduce. You have more than likely been overexerting your whole playing life. The Singing C Series addresses this exact problem.
Go back to the 1% Rule and the idea of humming, singing and body resonance and really relate it to the process of playing.
I hope that makes sense and helps.