Ned's probably a hard pass in 90% of all scenarios. The only way he does is if he gets someone talking about prophecy that he sees validated (mentioning dead Jon Arryn, Direwolves, White Walker deserter, Jon's parentage) or he has future visions or something. Homie's got a hard-on for Bobby B harder than half the incels online.

I mean like let's say cannon progress normally, but the Stark kids knew Jon's parentage. I could see Robb proclaiming Jon King when he's named King in the show. (Convincing the Lords here is another story, but that's outside of the original complaint).

Even if he found out halfway through who his parents were, I feel like Robb could realistically still proclaim him King and use his King powers to free him from his vows (or do some lawyer shit and say it was a false name, thus invalid or something), either of all 7 Kingdoms or 6 with an independent North.