It solves some of the problems – a King Jon would probably not want to revenge himself on the Starks in the name of his father – but not all of them. For one thing, this doesn’t exactly do much for the Tullys, Baratheons, Arryns, and Lannisters, who don’t have the same protection of blood.
But there’s a larger political problem: the rebellion was launched as an argument that the actions of King Aerys and Prince Rhaegar were so contradictory to the feudal social contract that they had forfeited their right to rule. Naming Jon as King goes back on that in a really egregious fashion: by acknowledging Jon as Rhaegar’s heir, you’re accepting his abduction of Lyanna as legitimate, which the Baratheons are going to have a problem with. Moreover, as Rhaegar was Aerys’ heir, you’re accepting that Aerys was a rightful monarch.
Moreover, the Rebel Alliance had already acclaimed Robert as their King. So now Ned has forsworn his oath to Robert to put his nephew on the throne, which is going to be seen as both an unspeakable personal betrayal and dishonor for Ned and as rank self-interested treason in the eyes of the political class. Ned’s now going to have to deal with the Baratheons and the Arryns and Tullys who’ve sworn their oaths to Robert and his heirs, and he’s going to have to deal with the Lannisters who have every reason to fear a Targaryen restoration.
So the only scenario I see coming out of this is a bloody and entrenched civil war, likely leading to the fracturing of the realm as the Vale, Riverlands, and Stormlands declare for King Stannis, Dorne and possibly the Reach splits the Targaryen loyalist faction (because the Dornish are not going to recognize anyone of Lyanna’s line ahead of Elia’s), and the Westerlands and Iron Islands are all out for themselves.
And since Ned’s not an idiot, he’s going to see this coming the moment he finds out about Jon’s birth and that’s why he would never name Jon King.