Elsewhere on this board, I've shared my experiences in how I fell short on a concducting course in 'person management' skills which are also vital to any conductor.
I think if you're a conductor of a lower section band which wants to improve, then the MD of that band needs to have a working knowledge of theory and harmony (or 'construction of chords') as well as the brass playing skills. Nowadays, I think a working knowledge of percussion and the techniques required is also a boon! Also, (one thing that does get overlooked by 'valve' players - it was certainly something I was guilty of in my early days) is knowledge of trombone slide positions, with or without plugs. Also, cornet and horn players going into conducting should learn about the 4th valve on a euphonium, bass (and, in these days, baritone).
The knowledge of chords/harmony/theory etc. is vital to lower section band conductors certainly, as many a time when a test piece comes around particularly, where queries are made by the band as to whether they've got the right notes in a certain passage. In these days when the music is printed from packages like Sibeluis, you tend to find that mistakes in the parts are replicated on the score and vice versa, so the MD really needs to be on his/her toes in the harmony department.
I can't comment on what it's like for top section conductors as I've never conducted one, though I'd imagine if I ever did, most of the band would know at least as much and probably more about the brass technique things certainly. However, there is a saying in music that the best players don't necessarily make the best conductors.
I've been conducting brass bands/ensembles, wind bands, orchestra (once) and choirs and whilst I'm confident I can pass on what I've been taught over the years, I know I've got a hell of a lot of learning to do. That's the challenging part and it's what keeps me going and wantinng to continue to conduct.