I have a hypothetical situation in my head. Picture a festival stage, lots of bands and crews coming through. A band is on stage with their own FOH guy mixing out front but they're working with the festival monitor guy. One song in, the lead singer's wedge drops out... oh boy.
Ideally the guy would finish the line he's singing, gallantly cross the stage to another mic while shouting to the monitor guy what the problem is. With a little quick thinking that mic will be hot and have similar settings to the one the singer just walked away from and the audience is none the wiser. Even better they might get the impression that he's working the stage to get closer to the crowd. In a couple minutes the problem with the center mix is worked out, the monitor engineer or stage tech gives the singer a nod and the show goes on without any appearance of there having ever been a glitch.
That's in an ideal world. Lots of times things are so hectic that information gets missed and that same monitor guy could look up from getting shouted at by the next band's manager to see a very cross singer standing at a muted mic. Beyond the simple mechanics of conveying information in a loud and chaotic environment, there's a lot of room for ego to take over and that's what we want to avoid.
I've been through quite a few foul ups where the artist winds up cursing out the sound crew on the mic or the sound guy does the same about the artist in front of a bunch of fans. But I've been through a lot more where even when the show ground to a complete halt, the artists just struck up some banter with the crowd, or pressed on despite the difficulties.
That's the kind of attitude that makes for great shows. I've had artists come off stage and say with a smile on their face that they didn't have a stitch of monitors. But they just grinned and played and the crowd had a wonderful time. That's gold. In fact, it can really be a way to pull the crowd in to the experience. Not every artist is trying to have a show be like hanging out in their living room. But even for an act with a more, I don't know, show-type-show, a few seconds where they solve a problem in front of the crowd can really make everyone in the building feel like they're part of the show.
I'm going to cut off before I start to ramble. Just keep in mind that attention to detail is everything and doing your prep work to get in sync with the artists you'll be working with can keep a mole hill from turning into a mountain.