As a key opening is increased the tone becomes less "stuffy" and the pitch goes higher. This change takes place until the opening is approximately 1/3 of the diameter of the tonehole. Raising the key more than that distance has no further effect on the tone or the pitch. There are some who claim that giving the keys an extreme opening increases the maximum volume and "projection" possible on the instrument. I can't comment on that since I don't play music that requires me to play as loud as the saxophone can play for an extended length of time.
I don't subscribe to the notion that on some saxophones lower keyheights make the instrument play/sound better. This is contrary to the idea that each note on the saxophone should speak clearly and with an "open voice". I once measured the tonehole diameters of the stack keys on a YAS-23, took 33% of that distance and compared that to Yamaha's recommended key heights for those keys. There was significant correlation when comparing those two figures. I take that to mean that the designers and acoustic engineers at Yamaha have a good understanding of how to "set-up" their saxophones based upon sound acoustic principles.
The more I learn about the acoustics of the saxophone the closer I come to understanding how complex all of the various relationships that go into making a great playing instrument really are. When pulling the mouthpiece out you are not only making the "sounding length" of the saxophone longer, you are also adding to the "effective volume" of the mouthpiece which needs to be a close match to the volume of the "missing cone" to its apex. The distance the soundwave travels to the leading edge of the tonehole and back is what partly determines the pitch of the note, but so does the temperature of the air, the "end correction" at the tonehole, the mouthpiece input pitch, the diameter of the tonehole, and yes---the height of the key. When the mouthpiece is set the correct distance on the cork it "tunes" the fundamental of the note that is being played. However, if moving it to that location causes the "effective volume" of the mouthpiece to no longer be a close match to the volume of missing cone, the harmonics will no longer be "in tune". This is most noticeable at the 2nd partial or the octave of the note. This is why I believe tuning must include matching the pitch of "overtones" to the pitch of the regular fingering of that note.
The bottom line is that I think it is a really bad idea to try to "tune" a saxophone by raising or lowering all of the key heights.