I can see both Kara's and Lena's positions on the kryptonite issue. On the one hand, Lena's right -- there are thousands of things that can kill humans, many of which we still use and work with every day -- fire, water, cars, knives, all sorts of things. So for someone who can only be killed by one or two things having a zero-tolerance policy toward their existence does seem like an exercise of entitlement.
But on the other hand, the fact that kryptonite can't really do anything but kill Kryptonians means that, unlike fire or water or cars or knives, it can't really be anything but a weapon -- and a particularly brutal weapon at that, the equivalent of poisoning a human with plutonium, say. And it can't be a weapon against anyone but a Kryptonian, and those are pretty thin on the ground in the galaxy, let alone on Earth. So Supergirl and Superman do have some justification in seeing the collection, let alone the manufacture, of kryptonite as a threat directed personally at themselves and their family. There's not really much else it can be. Even if the counterargument is "Well, we'll only use it to kill bad Kryptonians," that's still kind of an assault on their racial identity, their tiny community of refugees. Their species is already nearly extinct as it is.
Now that I put it that way, I think I come down more on Kara's side than Lena's. I believe, of course, that Lena has no evil intent in creating kryptonite, but I do think she doesn't really understand how justified Supergirl is in feeling personally threatened and offended by such an action.
