In no particular order:
I agree with the overall idea that Supergirl is not adhering to the code people might expect of her. However, in reality, there has always been some violation of morality where she's involved, and could scarcely be any other way. A superhero with a secret identity is already practicing deception to some degree, and this often involves outright lying, which is a violation of moral code to begin with. We ARE dealing, past Season 2, with a Supergirl whose moral affronts are at once more subtle AND more blatant, however. The show is no longer an all ages affair, assuming it ever truly was.
It's quite possible the writers feel they are exploring the character more in depth.
The lethality "works" to some extent regardless of what point of view you examine it from.
Kara the Kryptonian should think to dispose of White Martians from the point of simple logic. What graver threat is there among any species we've seen on the show so far?
Kara the Human has herself and her family and friends to protect. Emotionally, she should want these alien nightmares gone and gone forever.
Kara the soldier is combatant on a battlefield, with most of her allies down or weakened, and she herself is now alone facing enemies who not only rival, but possibly surpass her strength and speed, and who can take away the advantage she has if given even 1 moment to attack her mind.
There isn't anything I saw first run of this episode that led me to believe Kara had any reason to think her side was "safe" until she blasted away her opponents. The thing I found jarring was not the action, merely her cavalier attitude toward it.
I'm not 100% sure Kara hasn't killed even in Season 1 after gaining theoretical advantage, though.
Consider the 2:20 -- 2:30 mark of the following clip, featuring Kara's final confrontation with the Master Jailer. Looks and sounds suspiciously like a neck being snapped here, and, maybe I just missed it, but I don't recall ever seeing or hearing mention of this guy ever again ...