Similar to BCdotWEB, I'm a bit on the fence of whether there's an underlying valid question. However, you stipulated what part you specifically want an answer on:
Why do some people believe Cartman genuinely cares for Kyle?
It's a question in the sense that I want an answer as to why people think that, something that in a way might rebuttal what I've said
The short answer is this:
Because there have been instances where Cartman did something for Kyle that came from a place of wanting to help a friend. These kind moments also help prove that Cartman's usual hurtful comments are not the literal truth.
As much as Cartman can rip on Kyle and say that he wants him to die; at the end of the day, he chose to save Kyle in Smug Alert. That indirectly proves that he doesn't hate Kyle as much as he claims he does.
Yes, you're somewhat correct that this kind gesture is tainted by the fact Cartman misses Kyle as his favorite victim, not just as a friend.
However, it does still prove the point that Cartman's usual hateful anti-Kyle rhetoric is not true to begin with. It showcases the point that when Cartman rips on Kyle, he's exaggerating.
This ties into a larger part of Cartman's personality. Cartman has hateful opinions about certain groups of people (Jews, hippies, ...). But when you observe Cartman over all of the seasons, you start to see that his bark is much worse than his bite.
Cartman has learned to act offended when it helps him get what he wants. Cartman doesn't hate for hate's sake, he acts hateful because it benefits his true goal.
When Cartman wore the "Token's Life Matters" T-shirt, he wasn't trying to bring focus to racial issues. He was trying to bring focus to the fact that he is standing up to racial issues. Cartman's goal was self-aggrandizement; not social change.
In "Kenny Dies", Cartman attempts to get stem cell research (using aborted fetuses) legalized in a feigned attempt to save Kenny's life. It is revealed at the end of the episode he really wanted to use stem cells to clone his favorite pizza restaurant.
In "Ginger Cow", Cartman actually proves my point by completely pulling a 180 when it benefits him. Initially, Cartman keeps up the prank that the ginger cow is real. When confronted with an outraged Kyle, the school, the media... Cartman holds to the claim that the cow is real.
But then, peace in the Middle East is created specifically because the ginger cow is thought to be real. Cartman suddenly feels the need to come clean, but he confides that in Kyle, specifically. It's not all that subtle that what Cartman really wants is to extort Kyle, by making Kyle his servant in exchange for not revealing that the cow is a fake.
Cartman's actions change based on what benefits Cartman the most.
He first upheld the lie for personal amusement. He was enjoying it way too much to give up on it. But when he realized that he could instead extort Kyle (which would bring him even more joy), he changed his position so it helped him get what he wants (Kyle's reluctant obedience).
When Cartman victimizes Kyle, this is nothing more than a power play. Cartman wants to conversationally drag Kyle through the mud and retain social control over him.
As Cartman's advocate, I could argue that Kyle has a different-but-similar need for social control over his friends. Kyle is the one who most likes sitting on his moral high horse and behaves like the bringer of wisdom, even if others (sometimes including Stan) get tired of Kyle's harping on.
In that sense, Kyle and Cartman's treatment of each other is balanced. Kyle gets to act morally high and mighty (which most commonly consists of explaining why Cartman is being a dick), and Cartman gets to drag Kyle down and ridicule him (by trolling him).
Cartman and Kyle are two extremes at opposite ends. Based on a given topic, we can usually already predict what Cartman and Kyle's opinions are going to be. But Stan is the wildcard, and the more interesting character to observe. His opinion is not set in stone. He evaluates things independently and then makes his own choice.
- Cartman has no problem with amorality. He does what he wants. To Cartman, there is no boundary that you can't cross if you want to.
- Kyle is the opposite. He often argues the point that there are moral lines which cannot be crossed at any cost. Kyle takes this too far sometimes, holding on to a principle to a degree that is not healthy or helpful anymore.
- Stan is the mediator here. He simply sides with whoever seems to be the most correct in this particular case. There are times where Stan actively agrees with Cartman. There are other times where Stan tries to stick with Kyle, but eventually has to concede that Kyle takes it too far. Other times, Stan completely sides with Kyle. But there are also times where Stan, while agreeing with Kyle, ignores Cartman and doesn't actively feel the need to prove Cartman wrong (which Kyle is less capable of).
Just to not exclude him: Kenny generally swings the same way as Stan; he picks a side based on the options in front of him.
However, if the plot requires the boys to split off in two teams of two each, since Kyle and Cartman are practically always on opposing teams; Stan and Kenny divide themselves to even out the groups. Most often, Kenny and Cartman group up, because Kenny has a "naughty" side to him that connects with Cartman's amorality.
Come to think of it, Kenny actually proves the point. Especially in earlier episodes, Cartman ripped on Kenny (for being poor) as much as he did on Kyle (for being a Jew).
But Cartman still likes Kenny at the same time. They often pair up. They genuinely have fun together. But when Cartman takes a joke too far, Kenny can (and will) lash out at Cartman. Kenny does have a thicker skin than Kyle, he doesn't get upset as easily.
Essentially, Kyle is a more sensitive Kenny. And because he's more sensitive, it's easier for Cartman to upset him, which is why we see Cartman/Kyle rivalry more than Cartman/Kenny rivalry (Kenny has no issues just walking away if he's fed up).