Noah Lambert: >It is becoming more apparent to me that the best embouchure
is the one you are currently playing with. If you don't have range or ect.
it seems to me that is probably because most people don't have good airflow.
(...)I was hoping that people who have gone through this can discuss why and
what benefits did they get.
I haven't addressed this in a while (since the topic causes huge arguments) but in my personal experience an embouchure change can be hugely beneficial.
My original embouchure wasn't horrific (I had a decent high C as a 9th-grader in '81) but I had my lips spread out when I played, and the rim of the mouthpiece was pretty deeply "in the red" of my lip.
As a result, I used much more pressure than I needed to, and didn't have much flexibility or endurance. My teacher, the late Delbert Dale, told me that he blames world wars on bad embouchures, and that I should try buzzing without a mouthpiece. I did, and produced a very clear tone (not a raspberry sound). He noted that this was not the embouchure I used to play the trumpet and told me to try buzzing while I brought the horn up to my lips and to begin playing a note on the horn without interuption.
He encouraged me to avoid overlapping my lips, and to get the rim of the mouthpiece "outside the red." After a few days of experimenting, I found an embouchure that worked beautifully and my range shot up several notes (which is quite a revelation for a 10th-grader). He was so pleased that he gave me a free lesson and promply nailed me for not practicing my Clark book.
I can say that the embouchure change was one of the most significant events of my life, since it made playing the trumpet so much easier. I began using a fraction of my origianal pressure, and as a result was (instantly) able to reach higher notes with the same effort, along with more endurance, flexibility, and better tone. The same thing happened to another trumpeter in my band, so this isn't a fluke, nor was it reserved only for players with truly horrible embouchures.
If, on a "pressure" scale of 1 to 10, you're already at "9" when you get to high C, you won't be able to get much higher no matter how you breathe; you're simply pulling back too hard. For me, the embouchure change dropped my pressure at a high C from "9" to around "4" or "5". As a result, I didn't reach '9" until several notes higher. Which is as high as anyone really needs to go, unless you're auditioning for the Tastee Bros. Most of the embouchure points that Delbert discussed with me can be found in Farkas' book "the Art of Brass Playing".
I've seen too many people who practice several hours per day who use tons of pressure and have many problems with range/endurance, along with many people who rarely practice and have plenty of range/endurance, to believe that embouchure is not extremely important. I've also seen a number of people pick up the horn for the first time in their lives and (with a good embouchure) blow a solid double-g or above and say "you mean that's supposed to be hard?" Air support is, of course, crucial as well, but I've seen a good embouchure instantly fix problems of pressure, endurance, flexibility, and range that could have taken months and years to solve otherwise.
I'll go out on a limb here and say that in my experience, many of the great players/teachers who teach breath support without mentioning embouchure seem to have begun playing with a great embouchure and possibly don't realize how much advantage it gives them. But I'd compare playing with a bad embouchure, to swimming with bowling balls chained to your legs. Imagine what it felt like to have those weights removed!
Lastly, let me comment that I don't constantly worry about my embouchure today; the problem was solved for me almost 20 years ago over a short span of time and is not an ongoing issue as I've heard some people claim it will be. I'm not an expert on *teaching* embouchure, but I can testify that changing it worked for me and quite a few people that I've observed. I hope this at least helps answer some questions.