'The air does not travel through the saxophone,'
That is totally wrong - where do you think all that air goes? Air is compressible of course but there's no stopper in the bell of a sax. In fact, if I forget to remove the bell stuffer before trying to play, there's no way to play beyond a certain point that is higher than you would think on the sax.It even comes out of the bell when all the keys are closed. I'm not saying anybody could blow out a candle playing sax but all my life I've blown air out of the lower stack keys on alto, tenor and baritone - I feel it on my hands all the time, and I actually do not have a high-capacity for air. In fact, I don't do well in respiration tests but I can blow two saxes at the same time - alto with a Guardala Studio and tenor with a Guardala King Curtis, and handle fairly long phrases at a high volume. My problem comes in with not being able to hold notes for a long time at high volume on any horn, but I'd rather be able to push the air than have a huge reserve. Of course, it obviously would be better to have both abilities! Plus, the lungs lose capacity as we age, and I am definitely aged! I find myself having to think about taking deeper breaths and taking them before longer phrases, particularly on solos.
Wishing to blow more air through the horn is not 'wrong', but shouldn't be a goal by itself - you will do it naturally as you get more used to playing against the resistance of the reed and mouthpiece, and when you feel like you are being stifled, you will seek more open mouthpieces. Its a natural progression for some players.