Not sure those stories would have held up when (Book)Jon turned out to be the spitting image of Ned, looking more like him than any of his actual children save Arya. And that was the best case scenario, the worst was him taking after his father -- although I've seen it pointed out that that might be why Ned allowed the Ashara Dayne rumour to flourish.
I'd also say that in most scenarios, but especially in one's like Ned's where he's not a skilled liar, the simplest lie is generally going to be the best one. The more variables and outside factors you introduce the harder it is to maintain the lie. And it should be pointed out that his ruse was successful, he accomplished exactly what he intended and not only kept Jon safe and secret, but for the most part gave him a very good life.
I think it does make sense for Ned to keep Jon close, especially because he's the last living piece of Ned's sister. I have a nephew, and you bet your butt I would claim him as mine if my sibling were dead and my nephew needed protecting, even if it infuriated my husband. I would imagine Ned loved Jon from the moment he saw him, and he may have wanted far better for him than could have been afforded if he'd been raised by the blacksmith, for example.
I also green with Ashleyn that Jon as a bastard was a simple story that accounted for Ned's interest in the baby and their physical resemblance. I think that Ned's honor is also why the ruse worked - if Ned Stark says he's fathered a bastard, pretty much everyone believed him. Ned's name has also opened many doors for Jon, even if he suffered at the hands of Catelyn (and, in comparison, lots of other people suffered far worse at the hands of their actual parents). Even in the finale, Cersei referred to Jon as Ned Stark's son. There's a social benefit in having a connection to the honorable Starks, and Jon was educated, trained in weaponry, well fed, kept warm, loved by at least one parent, adored by several of his siblings, and primed for a leadership role pretty much as soon as he reached the Wall.
One thing I haven't seen mentioned was a tiny moment when Sam was talking to Bran and Bran said that Jon "is not my father's son" or something like that, which means that at least in that moment he still saw himself as Ned Stark's son, as Brandon Stark, and not just the three-eyed raven. I was so happy to see a spark of humanity in him!
I thought she really was upset, because northern independence was supposed to safeguard the interests of the north. She told him not to go because he'd be expected to bend the knee, and then he's gone for potentially months and bends the knee just like she thought. The Starks had finally taken back their home, reunited, and reestablished themselves as the most powerful northern family. Now they're beholden to a queen none of them know anything about. I could understand her frustration, even if as a viewer I loves me some Jon and Daenerys.