Jon's entire arc in GoT is him being a Stark and being worthy of the name, of course he is probably Raehgar's son, a guy he doesn't know and probably doesn't appreciate, knowing that he is Lyanna's son would be much more meaningful for him If he accepted, he would still see Ned as a father, no wonder he had such a strong bond with Ghost

possible, but it has a lot of potential and how things are going, it doesn't make any difference

Moat Cailin