i think in the same ways annabeth is incredibly smart because of athena being her parent, she is smart in all the ways she had to prove herself. a major motif of her character arc is not being taken seriously, and even with her natural affinity for weaponry and gaining knowledge, she had to work three times as hard (another reason im happy w the pjo series casting)
but in all the ways she is lovingly or maliciously called a know-it-all or being criticized for her need to prove how smart she is, one of the few people throughout the series that understands her 'why' is percy. here's a kid who had spent his whole life getting kicked out of schools, set up for failure, and put on a pedestal as a bad kid before he even does anything. in every bone of his body he wants to help, and he does it the only way he knows how: putting himself in the thick of it.
annabeth is an outsider, has few friends at camp because of how she interacts with people. her need to share what she knows isn't just a way for her to prove she is smart enough or worthy enough to be there - her ambitiousness and throwing herself into a quest is evidence of that enough. every piece of knowledge she shares is because she knows she is valuable, and would rather die than not help someone when she knows what to do.
throwing herself in front of the poisoned blade meant for percy was a tactical move as much as it was an emotional one. she knew she would lay her life down for him for her own reasons, but also knew enough about his fighting style and battle to know that losing him would guarantee a loss. being a know-it-all and being desperate to prove herself doesn't line up with making a choice like that: knowing what will happen, having the knowledge and strategy and pure instinct to decipher that in a split second does.
annabeth never wanted to prove herself for her ego (despite hubris being her fatal flaw). she wanted to prove herself because nobody thought she could help.
uncle rick: “I hear your concerns that the show won’t be faithful to the books. As a token of trust, have a trailer that’s literally the opening page of tlt word for word”
Look, I didn’t want to be a half blood. Being a half blood is dangerous. It’s scary. Most of the time it gets you killed. If you think you might be one of us, my advice is turn away while you still can. Believe whatever lie your mom or dad told you about your birth. Try to live a normal life. Because once you know what you are, they’ll sense it too. And they’ll come for you. Don’t say i didn’t warn you
But you wouldn’t, if you were asked? No, no, it’s not that, it really just depends on certain factors, like a number of things, but most importantly the quality of the script and the role.