UB: another brilliant post, as usual.
I thought about including roadies and FOH mixers, but the band doesn't consider them musicians (even if they're better players than the poseurs on stage). Still, back when I had the luxury of having real roadies (not somebody's roommate who works for beer), they tended to be the most level-headed people in the enterprise. You could usually count on a roadie to be the only one sober enough to trust behind the wheel.
As far as FOH guys, I have mixed feelings.
A good live mixer is worth his weight in cocaine. A bad one is worse than useless. The problem is, pretty much anybody who can write on a piece of tape without getting marker ink on the console seems to qualify as a live mix engineer. I have attended performances where the mix was just awful, where mics were feeding back and everything sounded like a bass solo on one string - and the mixer's flirting with a groupie, oblivious. Yeh, babe, I'm with the band. They couldn't function without me.
A good engineer is a lifesaver. Musicians tend to be technologically inept, and need someone to suggest that instead of smashing that transmitter maybe they could just try a fresh battery. And then actually be able to produce said battery.
I'm waiting for Karyn to weigh in.