Part this, and part the fact that a lot of satire glamorizes its target too much.

Take Starship Troopers. The protagonist is gorgeous, sleeps with two gorgeous women, is a football star, is born rich but doesn't take advantage of it, is loved by his parents/stands up to them/is still loved by them when he wants to come back, lives in a world with all sorts of cool sci-fi gadgets, joins the military (heavily glamorized in the real world and in the movie), he Tells the Truth and is a Good Soldier (even his bad boy fight with his ex's new boyfriend is done The Right Way), and eventually becomes successful (earning the respect of the tough, all-knowing mentor) despite not being particularly good at anything. Any pain he experiences along the way can be brushed off as a stage in the Hero's Journey. On a superficial level, it checks a lot of very appealing boxes. Even the violence has glamour to it: it's big and loud and splashy, if a significant character dies they get a distinctive death and usually some cool last words, and when the protagonist is injured he's perfectly healed almost instantaneously.

Not doing this is part of what makes The Sopranos so good. It puts all the ugliness of the protagonist right up front: he's fat, balding, trashy, self-destructive, destructive to everyone around him, intensely hated by everyone he cares about at some point, intensely hates everyone he cares about at some point, murders at least two relatives, is shot by his uncle, is involved in the least glamorous front business possible (literal garbage), and is ultimately shown to be not that big of player (he runs a "glorified crew" in New Jersey) and not even that rich (gets in financial trouble over 200K towards the end of the series, sees no lifestyle upgrades for all of his advancement, there are notable contrasts with the far greater wealth of Hollywood people). Everyone around him has a bunch of overtly shitty facets, too, and they constantly make trouble for themselves with how incompetent they are. The violence is unglamorous: characters will just get unceremoniously shot while pleading for their lives, there's a lot of focus on the unappealing work of body disposal, when someone's beaten badly they're shown to suffer effects for a long time (a character getting paralyzed in a beating is a whole arc, and he sticks around for multiple seasons), and when the protagonist is shot it takes him multiple episodes to heal.

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.