CATELYN STARK DIDN'T OWE JON SNOW A DAMN THING
1.) It was Ned's decision alone to bring Jon Snow to Winterfell to be raised.
2.) It was Ned's decision to not tell Catelyn Stark the truth of Jon Snow's parentage. (Whatever that truth may be.)
3.) 1+2 combined means that Ned is making all the major decisions regarding Jon Snow and, so it follows, is taking all the responsibility for his life and how he is raised. Catelyn has been pushed aside in this matter and chooses it upon herself to take no involvement in his life.
4.) And let me just remind you that while Ned probably would have been very understanding if he had married Catelyn and she had been pregnant with Brandon's baby, he would not have been understanding if she had become pregnant by say Petyr or someone. That is not even a possibility for her due to their patriarchal society and Catelyn accepts that.
5.) While Jon Snow lacks for a permanent mother figure, he is not raised in the modern nuclear household where alienation from one parental figure (or it's closest substitute) would be devastating. He is raised in a medieval noble house where he has wet nurses, Septas, Maesters, and more attending to his needs. And they do. He's not dressed in rags, sleeping in the cinders, left to live a servant's life. He's raised as a member of the nobility with all the attended luxuries. He eats the same food, receives the same lessons (and does better than Robb at sums), wears the same clothes, and is surrounded by siblings and a father who love him. He is raised like a Stark would be and even introduced to the royal family as part of the Stark household.
6.) So again, even though Catelyn doesn't allow him to sit with the Starks when the royal family visits (and bastards are a touchy for Queen Cersei let's remember) he is accustomed to doing so which is why is surprised and disappointed by this. Still, he is allowed to go to the dinner and eventually meets some of the royal family. Catelyn doesn't lock him in the cupboard under the stairs and claim he doesn't exist.
7.) Jon joining the Night's Watch is not seen as a disgrace to the Starks. He is following in Benjen's footsteps and the footsteps of all other younger Stark sons who had nothing to inherit. If things had continued normally it's possible Rickon would have joined him in several years time. Moreover, it was his idea.
8.) When Catelyn says she doesn't want him at Winterfell when Ned leaves, it's suggested that he go to King's Landing an idea Ned pushes aside because there's no place at court for him. There's no place at Winterfell for him either. He is nearly grown and will not inherit. When Jon was about eight or so he should have fostered with another house - maybe one that didn't have heirs - or set on a track to become a knight or a Maester or a Septon or something. He was not. And that is Ned's doing.
9.) And that must be baffling to Catelyn. Ned would surely send her sons, the trueborn sons she gave him, off to be fostered or to make their way in the world no matter how young. He is already making such plans to do just that with Bran by taking him to King's Landing. And though I'm sure there was a place for Jon at court, Ned wasn't going to bring him there. From Catelyn's perspective, he can hardly bear to be parted from Jon even to the detriment of the boy's future. Coupled with his refusal to tell her about his mother and it must seem like he loved this woman and Jon past all reason.
10.) When really it's just a strong indicator of R+L=J but she doesn't know that.
11.) So basically the bad things Catelyn has done at this point amount to: a.) Not letting him sit with the family as he is accustomed to do so b.) Ask him to be sent away which, while gratifying for her, is also the logical thing to do since he isn't inheriting anything c.) Treat him the same way she does Theon Greyjoy - cool and distant d.) One time she said something mean to him after not sleeping for days. Horrible crimes, indeed.
12.) And on that Theon Greyjoy note, while I'm sure some fans out there would want Catelyn to accept Theon into her life with open arms, there doesn't seem to be nearly as many. Ned chose to bring Theon Greyjoy into their lives as ward and as a hostage, his existence a continual reminder of the uneasy "peace" they have with the Iron Islands. Why does Catelyn own Jon Snow any more than she does Theon Greyjoy? She did not choose to have either of them in her life, she did not bring them to Winterfell and she did not want either of them there. Neither are her blood, neither are responsibility. Why does she owe Jon Snow anything?
13.) "It should have been you." Sure, it was cruel but Catelyn was out of her mind with fear and exhaustion at that point. More important though is how Jon handles it. He gets angry and then brushes it aside and continues his visit with Bran. It doesn't scar him for life, he moves on. Unlike some of his fans.
14.) Jon goes off to the Night's Watch, as planned, where he is made a steward. Jon does not like this and that is very telling. Part of the reason he does not like this is because he sees it as akin to being a servant and he feels he is above that. Jon is so accustomed to a certain life, a certain rank, that he feels entitled to it and is upset when it is not received. He is only mollified when Samwell points out he is being groomed for bigger and better things. So while some of Jon's angst is about not knowing his mother and that missing sense of his history and about some of the shame that he carries for the circumstances of his birth and his confusion over how his honorable father could do such a dishonorable thing, some of it is because he's privileged but not as privileged as Robb. Seriously. This is the great drama of Jon Snow's so-called life. Sure, by virtue of being a Lord's by-blow he lives a better life than most people in his region - in their entire realm - and is reared in comfortable surroundings and raised to be an educated man and surrounded by people who love him but some people have better lives than him and he is envious. I'm sure someone like Gendry Waters would cry him a river.
15.) For fun, let's compare fandom's reaction to Jon wanting the best life possible and even feeling owed it to say, Sansa wanting to be a Queen or Daenerys wanting to take back the Iron Throne. Funny how they differ, isn't it?
16.) Basically, Jon Snow had a good life and Catelyn did not prevent him from having a good life. She did not beat him, she did not steal food from his mouth, nor she did force Ned to choose between her and their children and Jon. She just didn't take part in his life. And why should she? She had no say in his upbringing and has no responsibility towards him. He wasn't her child and she owed him nothing.