Daniel - hi there. Apologies for the intrusion but I am new here and this is my first post!
Your perspectives on the complexity of classical vs. jazz suffer from being eurocentric if not ethnocentric. Because your background is (I assume) 'classical', the variables on which you assess jazz are essentially 'classical' - conventional harmony, melody, rhythm. The issues are far more complicated, however, when one is looking at different genres of music such as jazz.
Tone and timbre become more important, for instance. A jazz musician, for instance, strives to sound different to his peers, searching for a voice s/he can call her/his own. Classical musicians are generally moulded a certain way and fit into a pre-determined model of what a player should be i.e. a 'right' way to perform. So sounding like Coltrane is not a good thing!!
Jazz compositions tend to modulate more than classical music and tend to exist on more than one rhythmic plain at a time. Jazz players tend to use polyrhythms a lot more - in some ways a single jazz drummer is more rhythmically sophisticated than a whole orchestra, even one playing Stravinsky (and, for the record, that same drummer is considerably less sophisticated than a 10-year old Indian tabla student :O).
The hardest thing for an uninformed listener to 'get' is the concept of building a solo, using different techniques to create tension and release that takes the listener on a journey or, to quote Lester Young 'tells a story'. In order to do this creatively and in real time, the skills required of the player are far greater than those required of the much rehearsed motor skills of, say, a classical pianist.
Classical musicians play a note and sometimes add vibrato or a relatively small number of other 'effects'. Jazz musicians can grab a single note by the scruff of the neck and rip its head off, use it to sooth the savage beast, slur it, overblow, add too much breath, hold back, attack it hard - the choices for a single note tend to be wider and more reactive. Jazz players sometimes paint is broad brush strokes, where the notes matter less than the effect (Evan Parker is a great example, as was Coltrane. Noone will ever approach Coltrane and say 'you missed that Eb in the fourth bar of your 27th chorus' :P)
Classical musicians operate in a pre-determined setting (i.e if you are playing the eight bar of the second movement of Bach's Double Violin Concerto you will know exactly what the other people in the room are doing). A jazz musician is listening and reacting all of the time - what is the drummer doing, where is the bass player going, is the pianist playing the fifth?). Its the difference between reading a speech and holding a conversation.
Most significantly, the range of many jazz horn players is generally a lot greater than that of most classical players (Sibelius goes into the red along way short of most competent saxophonists, trombonists or trumpet players).
The arguments relating to drug use are complex and I will not revisit them here but, sufficie to say, drugs are NEVER going to make anyone play better. Ever.
These are all generalisations I know but I am only trying to illustrate the point. If you look at jazz from the perspective your are, you will miss a lot of what is happening. Can I recommend Christopher Small's 'Music Of The Common Tongue', a great, intelligent read that explains my point much more eloquently?