Ok, I've held off on this thread but as a part time FOH engineer I've got an opinion on it:
The FOH engineer's main goal SHOULD be to reproduce the sound of the band while doing as little damage to it as possible as they solve the problems inherent in every venue, PA rig, etc. encountered. Period.
So who SHOULD determine the sound? Ideally the band has an additional member that is the dedicated sound engineer. He/she knows how the band wants to sound, including the individual instruments, and is dedicated to producing the best performance possible. He/she is on the band's side, basically, but also has the necessary technical knowledge to make the performance sound as close to the band's vision as possible given the acoustic constraints and problems encountered in each venue. That's how the "big boys" do it and how they get their performances in the big concert venues.
That's the ideal. If the engineer is provided by the venue, as many/most are, then frequently the ideal is not realized, only because the engineer may lack the training, artistic dedication, etc. to do so. But that's not really the topic as I understand it.
I totally agree, but how is the audio tech to know exactly how an unfamiliar band is supposed to sound? So it may not be that the tech lacks training, artistic dedication, etc; rather the tech may simply lack the proper context to understand the band's sound.
Here's a thought experiment. If you were to record a band then allow every member of the band to create a mix, do you think they would all sound the same? I suspect there would be a lot of disagreement.
Ask a bunch of drunks in a bar what they think and they will all tell you something different. Let's keep that in mind when bands get bent out of shape because their buddies, spouses, or dates tell them the mix sucked.
Also sound is generally not uniform in a club. It might sound great in one location and crappy somewhere else. So maybe all the drunks are right.
AFAIK "the big boys" generally bring in their own techs.
