That's silly. He's spent a great part of his life waiting to be put on this wall, and then spent a great deal of time being mocked by the old guard who call him weak. They've spent years of time training him to kill with this sword, and he's been told repeatedly just how ugly and dangerous it is, and how likely he is to die out there.
And then they put him out there with her, and she's already pretty much indicated her willingness to signal their position to a bunch of people who are obviously going to kill him if they find him.
I think at this point, I'm with the veterans. If he isn't ready yet, I don't think he ever will be. The idea that it takes this long for someone to "learn how to kill" is a myth circulated by people who haven't been there. Put someone in fear for their life for awhile, and even someone from a sheltered lifestyle will kill in order to survive.
Put a guy in Iraq, give him basic training, put him in a situation where he's facing people who are ready to kill him, and guess what? He'll shoot them. He won't stutter and cry and aim his gun at the ground, or deliberately miss at the less second. I look at this sort of thing in these shows and I just laugh.
I might buy the 'She's just a woman' argument if she didn't seem blatantly hostile, and his position were not so obviously desperate. But frankly, women are people, and he probably would have been notified at some point in his years of training that 'mere women' are capable of killing and frequently do. They are human like the rest of us, and assuming that she won't slit his throat in his sleep is a strange assumption for a man in his position. I watched that scene, where he went to sleep next to her, and I lost any respect I ever had for the author. This is too stupid for words.
I expect it in a lot of shows, but Game of Thrones is supposed to be a dark realistic series. But even the dark series are silly and unrealistic about these things sometimes.
I imagine that later they will try to convince me that Jon will magically become an utter badass, but I'm afraid that will be a hard sell. Nobody takes this long to grow up.