By the late 80's I was a professional sound person. Whatever that means. But I did get to work with some people who knew what they were doing. So I learned a lot. And one of the first big things I learned was: live broadcasts don't have to sound like poop. In fact, a badass sound mixer who really knew what they were doing could actually mix live and to tape and have the tape come out good. But boy that's hard (I never became that badass). For instance, you sorta have to guess how loud the drums should be on an auxilliary send (relatively speaking, and counter-intuitively, they should be louder than the vocals because the drum sound is so much louder in the room you don't have as much gain on them as you do the vocals.)
Now somebody at the Letterman show knows how to mix live bands. And it makes me wonder if there isn't some fascist overlord at the Letterman show who says to the band beforehand "If you jerkwads don't keep your stage volume down, Dave will kick you off his show and instead do a little bit of business with Paul for the length of time it would have taken you to play your song and nobody will ever know you were supposed to be on. So when you're old and alcoholic and have no career, looking back on your pathetic, miserable, wasted lives, you'll have no one to blame other than your stupid selves who didn't keep your amplifier at the position we marked earlier in rehearsal." But I don't know. I could just be making that up. Then again, maybe they aren't as polite as all that.
What I do know is the number one secret to mixing. It's the biggest single thing and everyone I've ever met who knows how to mix and get it right immediately does the same thing.
I'm talking about how loud it is in the room, not how hot the levels are to tape. Your sound pressure levels have to be low enough for you to talk over or you're just going to get confused and not have any idea what you're mixing anymore.
Now I'm not entirely sure that this Airborne Toxic Event is a live performance. I mean, well -- I don't see how the string players are monitoring. That would tend to indicate they're miming along to playback. But then again. Hmm... At the same time the strings are out of tune. That would tend to indicate that they are indeed playing live but not able to hear terribly well.
This Death Cab for Cutie performance seems fairly "live". That's not to say it wasn't mixed again some hours later after they finished performing it. There are advantages to mixing (a live-recorded thing) later on -- namely that you can mute any microphones (which are open and would otherwise be creating a reverberant "mess" of the mix) which aren't being used. I'm thinking the second vocal mic in particular. And notice that the lead vocal microphone is almost exactly aligned so that the drums will be as far off-axis of it as possible. Well, for that matter the second vocal mic is turned around backward...