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[Game Of Thrones] would things have faired any differently if Catelyn had loved Jon as a mother

In Season 3 episode 2 "Dark Wings, Dark Words" Catelyn admits to Talisa that Jon was sick and she prayed to the gods to heal him. In return she said she would love him as her own, and ask Ned to name him a Stark. She couldn't keep her promise and blames everything that had gone wrong on her mistake.

Would things have changed at all if Jon was a Stark and considered one of Ned's sons?

[I have watched the latest season so spoilers are fine!]
gpt
Had Jon been legitimized and felt like a full fledged Stark, he would be unlikely to want to have joined the Night's Watch.

If that is the case, many of the events leading to the Red Wedding may not have taken place. He would have been an effective check to Theon, most likely preventing his coup and the supposed deaths of Bran and Rickon. Though to be fair, Theon may never have had the opportunity to stage a coup because chances are he would have been marching south with Robb while Jon bolstered and gathered troops in the North.

Also a legitimized Jon Snow would make Robb think twice about breaking his marriage alliance with the Frey's because (at least in the books) one of his main reasoning for marrying Jayne Westerling was concern over their bastard child.