One topic addressed often by brass teachers is breathing. Breathing is really important! Many horn players don’t breathe well.
However, there is another point that could be made and should be made to some students and younger teachers: breathing may be a part of many brass playing problems but it is not the answer to every problem. At an extreme if someone for example misses a high note in a passage the default reaction is they were not using their air well enough. I have heard comments of this type too many times. What the commenter didn’t seem to understand is that in fact it could actually be that the player is just stiff or over warmed up or just chipped the note, which is rather easy to do on the horn. Ask any horn player, there are lots of way to miss notes besides not hitting the breathing well.
Again, while air is a component of many playing problems and great breathing must be learned to reach a high level on the horn, air is not the answer to every problem.