The air that enters the mouth has various properties including moisture content, carbon dioxide content, warmth, etc, but the only quality that has any relevance to "support" is its pressure. That is imparted by the abdominal muscles and the intercostal (between the ribs) muscles, and possibly
opposed by the diaphragm, whose actual function is to
inhale, not exhale.
It also has the property of speed, which is relevant IMO only if the flow passes a constriction that at that speed, causes the air to become turbulent.
It also has the property of
shape, which is highly relevant for reed players, but I doubt it has much relevance for flute/piccolo players unless it causes constriction - see above. After all, the sound of the instrument is created a few mm
outside the player's body, and the mouth as a resonant chamber is separated from that by a very small lip orifice with pretty fast wind blowing out of it. so how would what is happening outside the body "feedback"/"communicate" significantly with what is inside the body.
All in my analytical opinion, with some formal fluid dynamics education behind it.
