Every year after ensemble placement auditions at ASU thoughts will come to mind. I was really happy overall with the quality and results of the auditions, but as I reflect on them a theme has been low range power.
In short, you have to have some power in the low range and it is a topic that comes up with every student. I even put together a book on the topic, Ultimate Low Horn, with great materials to use in lessons.
Not to take anything away from my own teachers either, but lessons on etudes and excerpts are a great start but only get you so far. Actual experience playing a lot of low horn in good groups is what gets you there.
In my own case the place I really got my low range hooked up was during my Doctoral studies at IU, playing Fourth Horn for two seasons in the Evansville Philharmonic on quite a number of standard works like Tchaik 4. And I was a good low player by then, but what I remember most was finally completely sorting out my break playing these works, figuring out how I personally had to approach that range. Those two years were an essential piece of the puzzle toward winning Third Horn in Nashville near the end of that second season playing in Evansville.
In my teaching I find that this range is very individual in terms of technical specifics, and where you get the power to finally go will probably be unique. Some decent power in the low range is not optional, work on it, play in that range in ensembles, hook it up.