Things that tend to lead to success:
1) Director points out key signature, time signatures, any major changes, etc. in less than 30 seconds and is onto fingering along the next second.
2) Director is having kids actively finger along through the piece and counting where they're at, as well as pointing out tricky rhythms, notes, odd things, etc.
3) Director generally makes it through entire piece in first segment of time at least once. (Tougher with longer more difficult pieces - but that's part of why at the higher levels there's more time)
4) Director gives an approximation of at least the rhythmic melodic material to to the band in the second timed section. Pitch perfect sight-singing could be nice, but most of our kids our used to our 'singing' to them.
The biggest thing is KEEP MOVING! No kid cares if a piece is in such and such melodic minor moving to harmonic minor with a perfect authentic cadence at the end. Seriously, teach your music theory at home, in this time keep things moving and get kids to practice fingering through the piece as many times as they can.
I've seen some directors emphasize the fingering that the kids must nail by having them show it to them. That can eat a lot of time, but may be worth it to you in some situations I guess.
Most of the bands I've seen struggle have had band directors who got bogged down in the explanation period with things that don't matter and waste two or more minutes before they even begin fingering along. Don't waste any time in the period and you can get through the piece completely at least two times.