For a lot of Comedians the notion of music in comedy is synonymous with the word hack. I know this because I've gotten onstage with a guitar a number of times in the past and, before having done anything heard comics groan "great, another guitar comic." In fact, one of the first times I met the fantastic comic Hampton Yount it was when I'd left my guitar after a show for a minute, he chased me out of the room with it in hand and said "Wait! You almost forgot your career!"
It's interesting that the word hack is typically used to talk about things that are overused, when really the frustration that arises with musical acts relates more closely to the notion of a computer hack. Because the thing about music is that it taps into a pre-linguistic, non-rational part of the brain. That's why there are certain songs which we love but know none of the words to, and why there are certain songs we hate but can't get out of our heads (see the chunk from Radiolab's Pop-music episode, Earworms, as well as their episode on Musical Language). For this reason, comedy has never had, and probably will never have the kind of huge fanbase that musicians can get. Yes there are comedians with big fanbases, but any comic can tell you it's not the same as the pure unadultrated love that musical acts get. It's because a lot of comedy hits us on the level of language and understanding and "funny ideas." It includes taking a position, and positions are tricky. You can disagree with a comedian, but as Demetri Martin has said, "you gotta agree with a song."
Using music on stage is a way to win the audience's favor without having to convince them via the avenue of the funny idea, so it feels like a cheat. In a way it's like the way Seinfeld has talked about cursing in comedy as "cutting across the bases." Yeah, you made it home, but you didn't run the whole way around. -- That said, the lyrical use of profanity from such comedians as Louis CK, Lewis Black, Chris Rock, Sarah Silverman, the entirety of Martin McDonagh's In Bruges and let us not forget the wide world of Armando Iannuci's creations, most paradigmatically captured in this reel of Malcolm Tucker (Aka New Who's new Doctor). Oh and I think Carlin may have had a thing or two to say about bad words too.
But I digress. Point is, with a few exceptions I'll discuss later, music in comedy is viewed as being an unfair trick, which I think is unfair, and tricky. Because there are any number of other "audience hacks" which comedians use all the time to gain audience love that don't involve music. These include, but are no way limited to: The rule of threes, listing things, yelling or generally using a bunch of energy (aka "selling"), repetition, references, callbacks and broadly speaking having a lyricality to their performance style. All these are things which if used for no reason can get to be hacky. Hell, this is why people tend to hate on Seth MacFarlane, because he tends to use repetition or other comedy techniques without any clear reason (consider, by contrast, Stewart Lee, whose use of repetition seems to be invoked for specific effect and for me at least evokes the kind of call and response that you hear in prayer). Sure, take away Demetri's guitar and some of those jokes might not work, but that's because (for some of his jokes, though very few in the last special) the guitar is part of the delivery mechanism. It's not that the jokes aren't funny without the guitar, it's just that they might have to be restructured to tell in a different way without it. Take away Louis Black's ability to curse or Jerry Seinfeld's ability to go up at the beginning of a sentence and then down at the end and they might have to restructure some of their bits too.
Different Uses of repetition
Of course, it's easier to point to a guitar and say "aha, bad thing!" than to describe the kind of structural elements which are also not in and of themselves "funny ideas" that still lure the audience into liking the performer or their style. And there are fewer music comedians, and when something is rarer, it's easier to spot, or as Mitch Hedberg once said to Shecky Magazine when compared to Steven Wright “I love Steven Wright but as far as him being an influence, I can’t measure that. Let me say this… if I made potato chips, and I decided to pack them in a skinny can, people would say I was like Pringles. But what if I packed them in a bag?”
We tend to notice the things which stick out and each individual becomes more of a representative for the group as a whole. This is part of the reason why when people see a showcase lineup (where you often have comics who are younger and still working their act out) with 6 really funny male comics , 2 mediocre male comics, 1 really funny woman comic and 1 mediocre woman comic they end up leaving going "women aren't that funny." And then when I point out that the other girl was really funny you say "well yeah, I mean she's one of the good ones." Well sure, in your sample size you saw a bunch of hilarious guys, but half of the women you saw were just ok, so she's going to represent more women for you until you see enough counter examples to realize "oh wait, I don't know what the hell I'm talking about." -- Throw in some societal stereotypes, a dash of confirmation bias and the fact that from a sexual selection standpoint men don't tend to care if women are funny so there's no social pressure for funny women the way there is for men and you've got yourself a terrible argument for why you shouldn't buy tickets to see Maria Bamford (There are a bunch of other reasons for this artifact, some of which were recently covered by the excellent Jen Dziura).
Similarly, you see fewer musical acts, so when you see one that isn't amazing you may think that indicts all of music in comedy. And then you see Reggie Watts, Flight of The Conchords, Steve Martin, Demetri Martin, Zach Galifianakis etc... and go "well, they're one of the good ones."
As a sidebar, Reggie in particular isolates the "sound of funny" where he delivers often meaningless sentences but referencing all the tonal and rhythmic parts of what make a bit work, in a sense giving a master class in the distinction between content and delivery mechanism. This is analogous to Andy Daly's bit from Comedy Death Ray where he provides a whole comedy show based just on hacky language, or Jeffery Joseph's bit he does entirely in spanish to a non-spanish speaking audience (I don't have a recording of it, but lord almighty if you're in NY see him live).
One other thing which I think affects comedians moreso than the rest of the population is this: comedians often have hyperactive brains and a somewhat detached, ironic or cynical worldview. This isn't always the case of course, but a lot of us are more comfortable engaging with negative feelings and finding surprising ways to be positive about them or doing the opposite with positive feelings. For many of us it's hard to turn your brain off, and it's hard to stop making clever snap judgments in every direction our head turns till someone snaps our neck for being so judgy. Music can be a way to get away from the noise in our heads. It can be a place where we engage emotionally, a place we can feel really happy without having to make a joke undercutting it, or a place where we can feel really sad without an aside about how in a 3rd world country someone is currently having their head bashed in, so buck the fuck up. I think for some comedians it can feel like bringing comedy to music is missing out on what is, for them, the best part of music. And, since any given musical comedian you're going to see is not going to be "one of the good ones," it probably isn't going to be your perfect ideal of comedy. And I have sympathy for this view. But I don't think that it makes sense to generalize from the subjective to the objective. It's one thing to say "I don't like it because of facts about who I am and how my brain works." It's another thing to say "it's hacky," "it's lazy," or "it's bad."
I like comedy that wrenches my gut, and I like comedy that changes the way I see an every day thing, and I also like comedy that I can tap my toes to. I think there's room for all of these things (as long as they're not women because obviously I just said that part to get laid).
All that said, one of my favorite bits of non-musical comedy is about music, and although I totally disagree with the point he's making, I'll leave you with the audio track from Paul F. Tompkins' amazing bit, "Jazz is Lousy," and below are a few videos of musical comedy, comedic musicians, and anything else that came to mind that I myself rather enjoy.
Enjoy. Or don't. Your call (he said, like a passive aggressive jewish mother).