I have always thought that finding out he was not Ned's would be a blow to Jon and not a boon. Plus, I think that Aemon's words to Jon were/are prophetic. Even if Jon rises to sit on a throne, he will derive little joy from it - much like his command of the Watch.
Although a thread over at Westeros was pretty hilarious - I think it was called what if X character was Jonned. (They all ended up skipping through fields of daises, and that is not how I see Jon's story at all.)
I cannot think that finding out he was another man's son would be anything but devastating for Jon. All of Ned's children loved him and he was a great father. Outside of his loyalty to the Watch, being Ned's son is Jon's driving force. What is important to him is being a Stark. I don't think finding out his is a Targ is going to be this great thing for him. Rather he will be saddled with a family he does not love and cares that he probably does not want.
When he decides to out and out break his Watch vows, it because of Arya and the threat the Boltons pose. They are threats to the North - threats to the family that he loves.
I also gag a little when it is assumed that Jon and Dany are made for each other. Hell there is little to suggest they would even like each other. She says one word about Ned and I could envision Jon tossing her off the Wall. They might become allies or even friends but I don't see them as this great love story. I may be wrong, but most people that were in it for love in ASOIAF ended badly. I would think that hoping Jon and Dany end up making doe eyes at each other would ensure their horrific death and destruction.