@Anna C, I don't find the method you've described nearly as effective - not that you shouldn't use it if it works for you and for your students, of course. If I could make a similar analogy, it would be something like cleaning a kettlebell, relaxing in the rack, then tensing before pressing. Perhaps not the perfect comparison - maybe relaxing the shoulders then packing and doing a pullup. Both of those things can and do work but I don't find them optimal.
For a musical instrument, the focus isn't on breathing in against the tension but breathing in fully and relaxedly. I teach beginning brass players to focus on getting their belly to expand as they inhale. Intra-abdominal pressure isn't an issue because all that's really of concern is taking in as much air as is needed. The pressure is managed on the other end by the vocal cords for singing or the embouchure (lips, mouth, tongue) for wind and brass playing, and part of the skill is balancing those two to achieve the desired musical effect. Control and strength eventually come to both ends of the chain, to releasing the air as needed, and to controlling the air at the other end.
-S-