Not sick.
the girl never understood/refused to accept reality as far as what was going on with people who died and were reanimated. she thought that it was possible to reason with them. she killed her sister to try to prove that point. no amount of trying to get through to her about that the people weren’t really there anymore, worked. she refused to see - denial too strong, perhaps a defense mechanism, but a fatal one where she underestimates danger and was getting angry that other live people were dispatching walkers.
when she turned the gun on carol and the other guy briefly, she became an overt threat ot the living people trying to stay alive. she just wasn’t getting it and was not going to get it. because of that, she became dangerous for the remaining living. she killed one, her sister. she drew a weapon on others. nothing was gonna bring her to reality. it would’ve been liveable if she wasn’t dangerous to others.
They could never trust her again. They would never be able to sleep around her again. Even if they did trust her with not killing them outright, she could have opened the door for a walker or allowed a walker access to them somehow.
She was mentally ill, and they had no other choice. If they had just left her, she could hurt some unsuspecting person. No other choice.