I don't know about the filling the horn with air concept. That one never made sense to me. Sure, baritones require more air, but I would think it is a function of the size of the mouthpiece and reed more than the size of the horn. If you put a soprano mouthpiece on a baritone, it will be out of tune and also take less air. Same size sax, but the mouthpiece would "take less air," not the saxophone. With a long enough neck extension, it might even be possible to play in tune. Putting a baritone mouthpiece on a soprano and it would "take more air."
The reed certainly opens and closes faster the higher the note. In theory, the reed could be open the same amount of time even though vibrating faster, but I don't think that's what happens. My theory on why higher notes require less air is based on looking at the reed like a diving board. What you do to adjust the "frequency" of the diving board is to adjust the fulcrum. Move it back and you get a long springy board with a slower action (the low notes). Move it forward and you get a faster frequency (the high notes). Moving the fulcrum isn't as obvious when blowing through the sax mouthpiece, but that is what is happening in order to change the frequency. You simply can't play a low Bb and a high F using the same length of reed.
The length of the air tube in the sax helps the reed vibrate at a certain frequency, but you can overcome this. Play a 2F and, keeping the octave key pressed, use your embouchure to drop to a low F. You will find that you have to loosen up and let more of the reed vibrate. If you take more mouthpiece in, you may get beyond the area where you can (almost unconsciously) control the vibration rate. But somehow you have to get more of the reed to vibrate to drop the pitch.
Air doesn't come in just over the tip of the mouthpiece. Probably a larger percentage comes in over the rails (at least on low notes where a longer length of reed is in play). The higher notes use just the tip of the reed and most of the rail stays sealed. That's why less air.
Sorry about the diving board analogy. I couldn't think of a way to use my usual car analogy.
Mark