caroline bingley is a good character for explaining the difference between an antagonist in a story and a villain.
an antagonist in a story is someone who stands in the way of the protagonist (not always the same as the main character! but in P&P it is), or at the very least is able to obstruct them, or something else. but caroline is not a villain because after darcy falls in love with elizabeth, she holds no real power. not even influence! she does try to sway darcy away from elizabeth by continuing to make jokes about her and her family, and while it can't be said that darcy doesn't disagree about the family in general, he is very quick to show his colours regarding his feelings for elizabeth. this might and probably does put caroline on the back foot, but she wises up relatively quickly, especially once she sees their behaviour at pemberley. caroline is an annoyance to elizabeth, and hurtful to jane, but she's not a villain. if darcy's opinion of elizabeth had never changed and elizabeth fell in love with him anyway, caroline might have had a villain opportunity then, but his opinion did change and it shuts caroline out of that role completely.
a villain in a story is someone who actually has power to derail things for the protagonist. to use a different example, what would change about the wizard of oz if the wicked witch of the west had lost all her powers, or never had any in the first place? without the ability to affect dorothy's storyline in any meaningful way, she is no longer a threat. caroline was never a real threat to elizabeth, and i believe this is true even before darcy falls in love with lizzie. i don't think miss bingley was ever really a contender to become darcy's wife, at least, not from his point of view. there would have been no point in darcy's waiting to marry her if she had any real shot at it, because we know from the book he can act quickly and decisively if he wants to.
this is why wickham is the true villain of P&P, and the only villain of P&P. it is his destructive impulses (and perhaps liking the idea of revenge) that nearly leads to the bennet family's downfall. the worst part is, they're not really his target. of course, he probably didn't hate the idea of getting to hurt elizabeth in the bargain because she was no longer taken in by him, but his actual target was always darcy and darcy's money, money wickham felt entitled to. the bennets were merely collateral damage in wickham's quest to try to inflict as much damage on darcy's bank account and his reputation and his social standing as he possibly could.
it's also why jane austen is such a brilliant writer, because she's got this whole vengeance plot going on and it's not even directed towards the protagonist! much of it takes place outside of the events of the novel and is already well underway by the time it opens. did wickham benefit from lydia bennet's neglected upbringing and education? absolutely. but even if lydia had never been in the picture, wickham would have still done what he did.
caroline bingley and lady catherine are definitely antagonists to elizabeth's protagonist, but they are not actually able to inflict harm in the way a villain can. i'm not saying they don't try, but they're not villains because they do not realise they don't have as much power as they think.
caroline bingley deserves better because she is not actually a villain; she does not deserve the punishment readers and fic writers love to inflict on her because though she isn't friendly to elizabeth, that is not an offence worthy of what often happens to her in other stories.