Air and...more air.
Air, through basic physical properties, makes it much easier for us to control everything about brass playing we do. Furthermore, it gives us something to focus on that keeps our brains from getting in the way -- just like a follow-through in tennis (or any other sport, I suppose). Air is the single and amazingly most predominant physical nessesity to play the horn.
This is where I begin my discord. That sheer physical ability can be a hurculean strength, and one that must be tamed; not through muscle control, but through a complete mental focus on the musicality you wish to achieve. As such, I am sorry, but I believe that at least 90% of what makes a good musician happens off of the horn. To get a good mental idea, we need to listen to a lot of music, both good and bad. All the time. A lot. Just as improtant, we need to get out and live life and experience those things that make us richer and fuller people -- and, hence, richer and more complete musicians (hard to play a love somg when you've never been in love, eh?). Finally, let's not forget that our horns actually make playing our mouthpiece a lot easier than it actually is. Spending time buzzing on the mouthpiece along is very popular among professional musicians as a consistent part of their practice routine, along with other non-instrumental things such as singing.
All of our technical issues can also be worked out on our mouthpieces. Articulate it on a mouthpiece, and I guarentee it sounds 10x better on your horn. And let's face it, people: we are not play clarinets or pianos, and we only have 5 (or 6) buttons, and they are always in the same place. You can teach any ape the fingerings.
As a teacher, the majority of problems are not solved by redundacy, but by breaking down the problem -- no matter how small or large -- and working on the fundamentals behind it. As such, the more time we spend off of our horns, the more consistent we will become. With that kind of consistency in place, brought about air and mouthpiece buzzing, we will have the tools to set out the musicality that has maturedin our minds based upon our listening and other practice.
Practice makes perfect, but remember, not all practicing is done on the horn. Besides, as an OK

tuba player once said, "I prefer to think of myself as a musician who happens to play the tuba."