This is just a thread made to show that if Jon were the one leading the armies circa AGOT, things would have been much better for the Stark family
1) No poisonous Tully influence
We all know what great contributions the Tullys made to Robb's cause. In that they screwed things up time and time again, releasing one of their most prized captives and withdrawing men from the Twins for example. If Jon were the one at war, he would not have to deal with this family at all, or the Riverlands. This would have made him more successful than Robb ever was. He could probably even get them on his side regardless for a common cause, but he would not be attached to them by any blood obligation
2) Jon would have supported Stannis
I have no doubt in my mind that Jon would not put a crown on his head. He would have bent the knee Stannis just like his father because he was Ned's truest son unlike Robb
3) Jon wouldn't trust Theon or Roose Bolton
Jon growing up as a bastard knows how to judge people's characters better than any of his trueborn siblings. He would know just how rotten and false Theon is and would keep him by his side instead of letting him free. I think he would have also gotten Balon on his side with a better envoy. He would have also never given Roose control of army as he knew Ned never trusted the man.
4) Jon would have traded Jaime
Unlike his brother, Jon wouldn't have abandoned his sisters in King's Landing. He would have traded them for Jaime and gotten them safely home.
5) Jon would have kept his oaths
Jon, like Ned, only besmirches his honor in times of need or when he needs to do so to save a family member. He would have kept any marriage arrangements and not be incredibly irresponsible and throw them away for some random fling
6) Equal education to Robb
We know from various recollections in Jon's PoV, that Ned taught Jon alongside Robb some lessons in how to rule. So Jon lacking education since he was a bastard would not be a problem at all. In fact, I would say from his time as LC he took the lessons to heart more than any of the Stark kids. He would have proven just as able (or perhaps more capable) as a military commander. And he would not have Cat around to give him some "smart" ideas
So all in all, I think that if Ned got Jon legitimized as a Stark before the events of the series, and he was the one in charge, he would have done quite well for the Starks.
Only person who would have stood in the way of that is the old bat unfortanately