lydia is a pale imitation of cordelia
I've already touched upon this thing before (where I included Caroline in the analysis), but since I'm rewatching BTVS, I figured I'd go into a little more detail.
Upon the rewatch, I realised that the main difference between the two is that while they both dish out slights and insults against everyone around them, Cordelia makes up for it as early as Season 2 because she's an active part of the group and actually throws down for them. She'll whine and complain about the monsters, staying behind school, but she's still pulling up in her car and offering Jenny and Willow a ride, she's out slaying with them when Buffy is gone, she'll stand in front of Buffy whilst she's unconscious and there's a vampire heading to kill her etc.
We also see the cracks in Cordelia's facade pretty early on, when she admits to Buffy that she's lonely. She could be surrounded by people and she's still lonely. We see that her popular group of friends only accept a version of her.
And honestly, Cordelia's bitchiness towards the main group can be forgiven sometimes because Willow and Xander aren't the best people (Xander moreso because Willow only starts to suck post Season 2). They're cheaters, they're hypocrites, they're bad friends sometimes. Her comments towards Buffy are obviously rooted in envy and because it's obvious, I barely bat an eye -- same as Buffy.
Even things like her saying she's good at standardised testing and that she has layers... all of it was implemented with Lydia Martin. And it failed. Badly.
Lydia is praised for her character development all the time and I'm not gonna argue that there wasn't some development, but it doesn't stick out to me majorly because there aren't any concrete moments unlike with Cordelia.
Cordelia has heart to heart's with Buffy. Lydia doesn't really get to have those with any of the girls, especially not Allison, her so-called best friend. The moment where Lydia's jealousy-induced cheater kiss with Scott is exposed to Allison, there could've been a moment where things slow down a little bit and Lydia admits her flaws and that she was jealous. She felt like her boyfriend was all over Allison and he treats her like crap, so she tried to make herself feel better by making out with Scott. But no, there's nothing. So there's no growth there.
Lydia is never, in my opinion, fully part of the group. Not in the way that Cordelia is. It's true that the vibes and the formats of the show are completely different, but it still stands that Lydia spent time with Scott and the others purely in a "saving people capacity". Cordelia went on double dates, she ate lunch with them, she watched Xander when he became part of the swim team etc. Lydia only followed the others in the cross country school bus in Season 3 because Allison was deathly worried about Scott and she was basically dragging Lydia with her. She only ever spends any time with the others because she feels the need to utilise her banshee powers, which is pushed on her by Stiles. In Season 3, we rarely see her unless she's being used for her relationship with Aiden or her abilities.
And even then, she was a reluctant participant.
She's like the groups rent-a-banshee, as opposed to a functioning member of their group who hangs out with them because she likes them as people.
Yes, there is some sense of "trauma bond" between Cordelia and the Scooby gang because she's one of the "outsiders" who was brought into the monsters and the vampires and she wasn't supposed to be, so she might have felt inclined to hang out with them more. But purely evident by her concern for them as people, bitchy comments aside, it's clear she likes them.
Any concern Lydia shows for the group is almost out of a sense of duty. They're a supernatural fighting group, not a real friendship group -- in my opinion, of course. Which is strange because like Cordelia, Lydia wasn't initially part of the supernatural stuff and was brought into it when she was attacked by Peter, so realistically, if the show hadn't isolated her in Season 2, Lydia should've sought out the group for answers and to feel less like a crazy person for what she had experienced.
Later on into Teen Wolf, it's a little more believable for Lydia to be around them in more casual moments, but there's next to no mentions of those moments. She spends time with Kira and Malia when Malia's learning to drive, in Season 5, and that's the only moment I can remember, really.
You could also compare Cordelia and Xander to Lydia and Stiles -- both pairs that should've never happened. And I think that if their fans had got what they wanted and Stydia happened way earlier, it would've ended up like their "counterpart", except worse because Lydia wouldn't have been able to admit that she's dating someone like Stiles. Cordelia had real problems with Xander and the public perception of it, but ultimately, she confronted her friends and told them to deal with it. Then we see her putting pictures of them in her locker. She was the one who got hurt in the end, the one that was cheated on. And I think if Lydia wasn't the one cheating on Stiles for someone she deemed better looking or more apt to be with her, it would've been Stiles cheating on her with someone who is easier to be with and who actually appreciates him.
Where they could've hypothetically developed a real foundation with Stiles and Lydia, somehow, and made it slightly more likeable and well-done, the writers just chose not to.
I'm not a Cordelia and Xander shipper necessarily, but I didn't despise them. She was the only person who seemed to tolerate him. He was the only person who seemed to tolerate her. It kinda worked. Even though it was toxic and worthy of eyerolls, it was a lot better than what Stiles and Lydia got.