A thorough presentation of the "Joe Allard Method" is difficult, because he did not really teach a "method".
Aside from a few basic ideas about breath support, embouchure, and the harmonic series, he taught different things to different students, depending on what they needed at that time (and depending on what he was learning himself, since he was always looking for better ideas). Every student's perspective of "what Joe taught" is different, because he taught the student, not a method. When I compare my lessons to my contemporaries, what Joe taught them was different than what he taught me.
Not once in the 3 years I studied with him did I ever hear him say "this is what you should do" or "this is the correct way to do X". What he would do is talk about various aspects of what he knew about how the saxophone worked, how the human body worked and how they interacted, or tell a story that demonstrated that interaction. This explanation or story would always have something to do with a problem I was having. He would then give me a suggestion of something to try. I would try it in the lesson. We would decide if doing what he suggested made it easier to play or made me sound better. It almost always did, but if it didn't, we would discard that idea and try something else. He would never suggest I do or practice something until he had demonstrated to me, in the lesson, that it made my playing better.
Of course, that is only how he taught me, and that was exactly how I needed to be taught. Which is why he was a genius!