The truth would legitimize Jon automatically and Rhaegar’s abduction of Lyanna though, because we both know at least are 99% sure they married. And I feel like you’re not taking account the personalities. Do you honestly see Hoster, whose tied to Ned by marriage, Jon Arryn who raised Ned like his son, and Stannis who had conflicting loyalties regarding House Targaryen and his brother, who would be out of the picture in this scenario, really challenging Ned if he proclaims Jon king?

Legitimize Jon, maybe (depends on how legitimate the marriage would have been considered. As we saw with Tyrion and Tysha, marriages can undone if deemed unlawful). Legitimize Rhaegar’s abduction, not so much. As much as it conflicts with our own values, Westeros is an extremely patriarchal feudal culture, and Lyanna had been betrothed by her father to another man and was not free to be married to Rhaegar (even if the very much married Rhaegar was allowed to marry her). As I’ve said before, accepting Rhaegar’s actions sets a dangerous precedent for every lord and landed knight in Westeros, that their ability to make dynastic alliances, and to decide the inheritance of their lands, could be countermanded at any time. 

Also, the personalities are more complicated than you’re giving them credit for: Hoster Tully had his daughter’s betrothed and his bannerman Jeffory Mallister murdered by Aerys II, and Jon Arryn lost two heirs Elbert and Denys Arryn to Aerys and Aerys’ armies respectively. And while Stannis did indeed have conflicting loyalites w/r/t to Robert, the legal situation is different now, because the rebel alliance swore a binding oath to support Robert and his heirs, and Stannis is Robert’s heir.