Hi musicteach - Please do share your take on why the Sax is so popular - I have an idea why but I need to hear from the pro in the trenches, ok?
Well first of all, it's a fairly easy to learn instrument, much like most of the woodwind family. It's a very simple design (again like most woodwinds) where it's one solid instrument plus the mouth piece. The only bit of tuning yay or nay would be moving the mouthpiece back and forth on the head joint. Again, that's for most woodwinds. Basically, once you learn one saxophone, you can switch between the other saxophones fairly easily without too much "growing pains". Plus it's considered a "cool" instrument. Not only that saxophone is one of the most versatile instruments. You can have them in classical arrangements, marching bands, jazz bands, rock bands, blues arrangements, it's really easy to have stand alone sax quartets. And really that goes for all of your woodwind family instruments. The clarinet is a little bit more picky in that it takes a bit more of a precise touch to get a good sound out of the clarinet then the sax family. The exception to this is the flute family. Flutes and piccolos (and bass flutes, alto flutes, and contra-bass flutes..) are a little bit more tricky because of the way the embouchure works. It's more of a whistle then putting your mouth on a mouthpiece.
But with all of your woodwind instruments, it's roughly 35% embouchure and 65% keys. Whereas on your brass instruments it's the reverse of that and more so. For your brass instruments, it's something like 70% embouchure and 30% valves. On a tuba, with three valves, you can still achieve the range of the entire saxophone family (provided you've got the embouchure). That's of course because of the harmonic series which is what allows us to play an entire series of notes without changing valve combinations, it's all embouchure. There's a harmonic series for each of the valve combinations (and slide positions on trombone) for your brass family.
And of course, I think we all know why percussion is so popular... Do I really have to comment on that? Your strings family are the most difficult instruments to learn, and thusly and the least popular.