I think you're onto something with the emphasis on power structures. A bit of off-topic speculation:

Jokes involving characters and a narrative often rely on status, and usually involve reversing a status relationship in some way. Classic example: Consider a person slipping on a banana peel. If the person is a struggling single mother in a poor neighbourhood, and slipping on the banana peel causes an injury that requires her to take time off work and lose income she needs to feed her family, that, to me, is not particularly funny -- she starts off in a bad and low-status place, and things just get worse for her. But if the President of the United States -- a guy who, at least in theory, is supposed to be dignified and respected -- ends a speech by saying "I promise that my administration will bring dignity back to America!" but then slips on a banana peel as he walks off the stage, that's hilarious. To paraphrase Krusty the Clown, a pie in the face is only funny if the victim has dignity to begin with.

But I get this sense that, in the US at least, there's not a universally recognized understanding of status -- possibly correlated with the lack of understanding of class and structural oppression (if only some Prussian philosopher over a hundred years ago could have described the kind of "consciousness" necessary to understand power structures in society). Which means there are a lot of people who just think watching a person get injured is funny in itself, whether it's the president or a struggling single mother. A lot of right-wing people struggle to understand the concept of "punching up" vs "punching down". Obviously I haven't done a study on this, but I bed you'd find that the audiences for various kinds of humour based around people getting injured without a structured joke beyond that are disproportionately watched by right-wing people, moderate to extreme. This would include e.g. Adam Sandler-type lazy slapstick, but also a lot of "dark" "humour" based around enjoying human suffering, "Darwin Award"-type enjoyment of stories and videos of people injuring themselves, as well as "edgy" comedians who claim they make jokes at everyone's expense but in practice tend to make jokes that support existing power structures.

Sometimes right-wingers are clever enough to realize that comedy works best when it's turning a hierarchy on its head, so they scramble to find a way to frame it as though the right-wingers are the underdogs. Which means that their humour has to attack high-status figures. They seem to have some success going after e.g. celebrities, Hollywood producers, bureaucrats, liberal politicians, university professors, journalists, anyone who's telling them they're being a jerk at any given moment and causing them to feel shame, all of whom can be construed to have high status. Or, if they want to be really insidious, they'll start out with self-deprecation to make themselves seem like the cute, goofy, innocent underdogs, e.g. pretty much anything involving identifying with Pepe or his variants.

Anyway, that's my half-baked theory of right-wing humour.