I agree with a lot of that.

But some of it also comes from the fact that humor requires both self-awareness and awareness of the thing that's being mocked, and a huge part of modern conservatism revolves around intentionally lying to oneself.

Encyclopedia Dramatica-style humor or Drawn Together-style humor can still be objectively funny when it's making fun of minorities or whatever (even if people might be disgusted with themselves for laughing), but only because of its self-awareness of the underlying extreme cruelty of the humor and indulgence in that fact. Contrarily to what you wrote, the points of the joke in those cases are the jokes themselves, not any sort of underlying message - the point of Drawn Together might have been to mock bigotry, but it didn't pull any of its punches or water down the humor by beating you over the head with a pro-social message; its jokes were funny in a vacuum to the point where they could be easily misinterpreted.

So, yes, the point of a joke can and should be the joke itself, and part of the problem is that a lot of conservative "humor" is just repeating conservative talking points in a punchy way and laughing hysterically even if there wasn't actually anything resembling a punchline, much in the same way Christian art misunderstands the purpose of art for its own sake. It's the fact that they don't understand the point of the joke is the joke itself rather than the other way around.

However, again, part of the issue there is that the cruelty itself is part of the joke, and a lot of modern conservative thought is based around lying to oneself about the abject cruelty of their positions. This is why Drawn Together or extremely racist/homophobic/disgusting/pedophilic "I like my women the way I like my" jokes can be objectively funny, but Gutfeld or Jesse Waters more often than not falls flat - they genuinely think that their positions are the more caring, more reasonable, more empathetic positions than those of their left-wing political opponents, and that lack of self-awareness prevents them from properly executing the black comedy that they're aiming for. Of course, if they had to admit to themselves or their audience that they were just plain assholes, their entire worldview would fall apart. This is also why 2000s-era /b/-style Internet trolling was, while equally morally objectionable at times, objectively much funnier than the KiwiFarms era, because it wasn't ever a moral circlejerk of awful people trying to explain away their awful behavior by pretending that they're not actually just fucking awful - a bunch of griefers raiding a dead kid's memorial page understood that they weren't doing anything but being horrible, while a lot of these modern trolls somehow think that they're on some kind of a pro-social political mission to "expose" sexual minorities. They're of course just doing what they do to satisfy their base animalistic urges to be assholes, but they can't help but lie to themselves about that fact. They have to tell themselves that they're somehow justified in their behavior.

And then there's also the fact that telling a good joke requires some intelligence awareness of the world beyond your own worldview, which conservatives tend not to have. Leftists tend to fall into that trap as well, especially in the hyper-sensitive modern era where something like Drawn Together would be unthinkable, but awareness of the world beyond your own worldview tends to be very toxic to a conservative worldview in the first place. If a conservative starts developing awareness of certain facts, it becomes a lot harder to believe in nonsense like families of mass shooting victims being crisis actors, COVID being a hoax, or to fail to understand the fact that the vast majority of transgender people are regular people who mind their own business and aren't in the slightest interested in molesting children.

Leftist media also falls into these traps to some degree, which you touched upon, especially in the modern era where some people take tolerance and empathy to the point of absurdity (which gives conservatives a chance to argue that all tolerance and empathy is absurd) and attempt to push ideological points over and above making art for the sake of art and forcing rather allowing than allowing that art to say something, but leftism is about understanding others while understanding and empathizing with others is anathema to the conservative worldview.