On Karen's purpose
Some people think the reason Karen Mccormick was created and always kept as a damsel in distress with no real agency was to give Kenny more focus, because he's harder to write due to being unintelligible most of the time but I doubt that it was the sole idea. Like, if they just wanted to make it easier to write Kenny, why not just make him intelligible more often, use his alter ego Mysterion more often (who is already intelligible), use his alter ego Princess Kenny more often, have him protect any existing character or one-shot characters that he meets once in his life or even just have him overcome his own struggles more like poverty or his deaths? Not only that, but ever since Karen was created she always had sporadic appearances. Contrary to popular belief, her first appearance was not "In The Poor Kid", it was years before that in "Super Best Friends" where he was background character and not even present for long, but just in a blink it or miss it moment. Why wait years to give her a role in a story? Why sporadic appearances if she's a shortcut for Kenny's character. Also, having Karen be proactive wouldn't have taken away any of Kenny's screentime nor been that difficult. Something as simple as her encouraging him for once wouldn't have been enough to justify her existence, but it still would've been better than her just waiting to be rescued and it still would've resembled a real relationship than what we got. Not only that, but after "The Fractured But Whole", so after Karen had three plot relevant roles and each of them being made once in a few years, they just pushed her into the background. I saw similarities between the bond Kenny and Karen share and the bond between Kyle and Ike. I commented on it in a previous post https://www.tumblr.com/medusasea/760641446053363712/south-park-younger-siblings?source=share (I mentioned there briefly that Karen's writing reminded me of sexism, but I highly doubted it was actual sexism in play. Now I am entirely convinced it wasn't about sexism at all). My guess is that the writers didn't just want a shortcut to give Kenny more screentime, but that they wanted badly to recapture the popularity of Kyle and Ike's bond on Kenny, because Kenny is massively popular himself, largely due to his heroism, so they thought that turning him into a big brother would make him even more appealing. However, since South Park already has a huge and diverse cast, they couldn't come up with a personality to make Karen stand out, so they reduced her to a passive damsel, sometimes trying to make her more appealing with some surface level traits, like pigtails with bows, a nasally voice, more dirt and changing her two pajama set with an onesie to make her more adorable and pitable, but when that failed to make her popluar, they tried a "rebellious" phase, when she suddenly joined the vamp kids despite Myterion's protests, then join the goth kids and they gave her a dark but cute makeover and even made her smoke and swear, but they kept her face dirty and she was still a passive damsel who needed to be rescued from the vamp kids, so an all style but no substance rebellion.
I don't have any proof this was the case and I am not sure this was the thought process behind Karen, but to me, this is the most likely explanation on why her writing is so melodramatic and superficial.