If I were interviewing a brass player, I might ask about clarinet B or Bb, saxophone F or F#, flute Ab. A WW player I would probably ask about trumpet F# or Bb, or tuba E or Eb. Or even better - "You're sight-reading a piece that moves from the key of F to the key of Bb. What notes do you anticipate your students will miss after the key change, and what fingerings would you remind them of?" That tests their basic theory knowledge (need to quickly know that E changes to Eb), transposition skills, and fingerings.
I'm not going to axe a candidate because they don't have the trill chart memorized, but basic knowledge is important to have in your head for quick adjustments in rehearsal.