If you hate the tropes so much, then why are you including them in your story?
No one is forcing you to write a story filled with stereotypical barbarians, Lovecraftian horrors, abnormally aggressive animals, and what have you. If you don't like the tropes, just don't include them in the story. In fact, the example you gave had no examples of any of the tropes you mentioned as being a pet peeve, and if you hadn't mentioned the character was intended to be a subversion I don't think anyone would have ever noticed at all. If you don't like stereotypical barbarians in your story why are you including them?
If you still have a functioning story, no one cares. The only way it would be a problem is if you broke the flow of the narrative to complain about how bad the trope is and how you are being different. If someone writes a story and all they do is complain about specific tropes in a non-humorous and vitriolic way, that story isn't entertaining, and people won't read it. Honestly, most of your question as written at the time of this answer comes across as more complaining about tropes you hate than your actual question, including pre-emptively refusing to understand and attacking anyone who tries to convince you why someone likes something, as you asked in the OP. That's the kind of thing you don't want in your narrative. It sounds like you are trying to pick fights with your audience, and that never works.
Building on this, who exactly are you writing for? From your question, it sounds like you are trying to write a story that draws heavily on pulp tropes...but has nothing but contempt for them. So, to clarify, you seem want to write an entry in a certain genre...while telling your readers, who likely picked up your story because they like the genre, that they are badwrong for liking it...and refusing to research the genre to understand the common pitfalls and create a more biting and accurate parody. See the problem here?
There's also the question of time. I like Hokuto no Ken, but it's
190-episodes long, so I couldn't get into it. Skimming through stuff I
actually dislike is going to feel like a waste of time, especially if
How am I supposed to understand why people like Jaws, Lovecraft, and
pulp so much? They're garbage as far as I'm concerned and I haven't
seen any prominent figure try to understand why someone would like
something they themselves didn't.
If you aren’t willing to do the research, you’re going to make a worse product. No ifs, ands, or buts. There’s a reason why “do the research” is a cardinal rule of writing. If doing the research makes you miserable, it may be time to question why you're writing what you are writing.
To use your Springtime for Hitler example, Mel Brooks knew the Nazis very well. He was a Jewish World War II veteran who fought against the Nazis in the European theater. Even though his professed life goal was to make Hitler and the Nazis so ridiculed they would never be taken seriously again, he drew the line at making jokes about concentration camps and the victims thereof because he knew it would be in bad taste. Because he knew and understood the subject matter he was trying to make fun of.
In fact, Mel Brooks is a pretty good counterargument. Despite parodying gothic horror (Young Frankenstein), westerns (Blazing Saddles), space opera (Spaceballs), spy movies (Get Smart), and others, Mel Brooks loved these genres, and as a result was able to parody them more effectively. Similarly, Neon Genesis Evangelion was written by mecha fans, Watchmen was written by someone who loves superheroes and writes superhero comics for a living (to the point that he laments that Watchmen appears to have made the genre too dark), etc.
I'm going to echo what the other commenters have said and say you don't have to like something to parody, subvert, or deconstruct it, but you do have to understand how it works. If you don't understand why people like something, it might turn out they like it for the same reasons you hate it, and don't like your work. Or they like it for different reasons than you hate it, and so your work comes off to them as a strawman argument.
As to why people like the specific themes or tropes you mentioned...
Pulp in General
The reason people like pulp is because it’s about a badass hero (of any gender) going on an adrenaline-fueled adventure in an imaginative and engrossing world, often with a colorful cast of memorable characters. It's cathartic escapism, and people love that. Tread very carefully if you wish to take people's escapism away, audiences hate it when someone tells them that they shouldn't be enjoying their chosen method of catharsis.
It is perfectly possible to write pulp adventures without any of the trappings of the early 20th century. A lot of the stereotypical trapping of pulp are more a product of the times they were written in rather than inherent to the genre. Adventure Time, if you think about it, is literally modern pulp. It’s about a badass hero who goes on adventures in a bizarre and unfamiliar world, with the help of wacky comrades and often to gain the favor of a princess.
The reason people like stories with abnormally aggressive animals or monsters or “always chaotic evil” barbarians or minions is because they want a cathartic action scene where the heroes get to punch bad guys in the face. They don’t want a thirty minute monologue on whether or not it is moral to kill the foe every time there is a fight scene, or a smash cut to every minion's grieving widow when they die. Musing on the morality of violence isn't a bad thing, but if it ruins the audience's catharsis that's bad. Making the foes unreasonably, irredeemably evil removes lingering doubts in the audience about the morality of the situation, because then they don’t feel bad when the villains get punched in the face.
Aggresive animals, in particular, have always fascinated us because throughout our history we have always been at the mercy of sharks, crocodiles, big cats, and bears. Humans have always been fascinated by struggles against large predators or "monsters", especially with the conflict of organisms that outclass us physically against human cunning. And the thing is, despite the fact that real apex predators are much more placid than fiction would suggest, a sufficiently hungry or angry shark, croc, big cat, or bear would kill people (not to mention an elephant or hippo), which makes the idea that characters have to deal with them more plausible in the minds of the audience. But from an emotional perspective it's no different from a story with a xenomorph, Predator, or Terminator, albeit with that extra bit of plausibility that something like this could technically happen.
The reason people like Lovecraft’s works (cosmic horror) is because of the existential dread and helplessness they invoke in their audience. In the time period Lovecraft was writing in, that was a very big deal because most people were religious to some degree and the idea that the universe wasn’t under the benediction of some benevolent creator deity was both unthinkable and existentially horrifying. The reason why people like Lovecraft has nothing to do with his racist beliefs. It’s just that due to the guy being terrified of literally everything, he was in a position to write stories which found terror in unexpected places. There are lots of good cosmic horror works that have nothing to do with Lovecraft. House of Leaves, Steven King, and the recent Annihilation movie are good examples. “Lovecraft” is not a genre, even though he’s one of the biggest names in cosmic horror. And Lovecraft himself has been parodied to hell and back because of how silly his stories are.
Indeed, despite these tropes being common to pulp stories in general, they aren’t universal and quite a few stories subvert them. The entire central theme of Conan The Barbarian is that despite being a “barbarian”, Conan is ironically more intellectual, moral, and civilized than the vast majority of his so-called “civilized” opponents, who have ironically degenerated into barbarism due to civilization making them soft and removing consequences to their actions. And in the Barsoom novels the “barbarian” Green Martians are portrayed as some of John Carter’s greatest allies, with one of the narratively richest, most introspective, and internally complex characters being the heroic Green Martian Tars Tarkas (and not in a "noble savage" way). This is exactly why you do research, to see the takes on certain tropes that have been done before and to see if a stereotypical depiction of a trope in a genre is real or if it's just a meme that's gotten out of hand.