I don't like a full blown edgelord Kara, but I definitely don't like it when she's just some "all American girl" either. And I absolutely agree that Kara and Clark should have plenty of tension between them. And not only just when she first arrives, but always. They're very different people in the way that only people who are very similar can be, if that makes any sense?
I think with Clark, Kara pops the illusion of Krypton he had built up in his head. Clark wants to think his culture was perfect, that his bio-parents were perfect, and then along comes Kara with her stories of how Uncle Jor would yell at his assistants and her casual anti-clone racism, and Clark has to face the reality that Krypton wasn't flawless. And of course, she reminds him of how very un-Kryptonian he is. She laughs at how he speaks the language and doesn't understand the slang (not in a mean way but still). Clark's always aware of how not-human he is, but with Kara he's reminded of how not-Kryptonian he is too, and he feels like an even bigger outcast.
And I think Kara resents Clark for not being more emotionally attached. He's the only thing she has left in the universe but he doesn't, and never did, think of Krypton as home and didn't know she even existed until he was an adult. He can't miss what he never knew and I think Kara resents, despite herself, having to mourn her world by herself. And I think she resents and is also jealous of how well Clark fits in on earth while she struggles to understand all of it, even basic body language. Clark thinks he's an utter outcast, but from Kara's perspective he's as human as anyone else on the planet and I think that causes a lot of conflicting emotions within her.