I know this is a route they like to go but I often find it kind of hypocritical.
Say both have to go stop a bank robbery with some armed robbers. You could argue that the only brave one in that situation is Kate as she is the only one who is any actual danger from gun fire. Kara is definitely being noble in that situation but I would argue in that situation she isn't brave. Bravery is to show courage even in the face of danger or pain and without Kryptonite or some seriously OP alien Kara has literally no threat of that.
Especially with how way over the top I think they have treated kryptonite. Like it being in the same city as Kara is an insult to her.
Okay... But, how is it hypocritical when the discussion is not about bravery but about vulnerability and fear? Kara doesn't say she is braver than other people. She says, she is unlikely to get hurt compared to humans. That is the truth.
She doesn't go into a situation involving armed robbery to prove she can do it.... or that she is brave. She does it to protect people. Because she can. So... has nothing to do with hypocrisy or even pride.
As for whether Kara is brave when there is a likelihood of her getting injured.. She has shown time and again that she is willing to go into situations to save people irrespective of danger; when she is powerless.
The Lena situation is completely different. The problem there was that Lena may not know this but to Kara, Lena is a friend. So, her having Kryptonite behind her back comes as a surprise to her. It'd be like your friend having something that could harm you and only you. I think their emotional entanglement muddies the picture for her.
Kara doesn't say hand over the Kryptonite when it's Lillian or someone else having it. She just goes in and tries to save the day. It is much the same way Clark reacted to J'onn having Kryptonite.
So. I don't see any hypocrisy. It seems more like a consistent characterization (especially when you consider that Kara doesn't say hand over the kryptonite to me but to Alex.. who had Kryptonite as far back as S1).
The only place where Kara does seem to use double standards is with Alex and her fellow DEO agents because while they are also consistently in dangerous situations, Kara doesn't seem to stop them from doing their job. I assume it is for the same reason that she doesn't ask regular law enforcement officials to just pack up and go home. Because she doesn't have the right (and with Alex, she was already a DEO agent for years before Kara joined up).
And... what does bravery mean to an invulnerable person? That is a whole other question altogether; but, I'd think it could be considered the ability to open their heart to love and friendship, hope and compassion, after suffering immeasurable and unimaginable loss... And, Kara has shown time and again that she is brave in that regard.