So what I just realized was that Joffrey, being the ignorant cunt that he is, would've left Winterfell without a second thought if Tyrion hadn't provoked his anger. Sending an assassin after Bran had nothing to do with covering his parents' tracks -- he was just trying to show Tyrion who's the boss, because of this moment right here. If Tyrion doesn't slap the shit out of Joffrey by the stables, then Cat never investigates what happened, Ned never gets suspicious, the starks and lannisters never fight..... maybe Robert still dies, and maybe Ned still investigates the bastards on account of Lysa's note, but I don't think he goes to war over Jon Arryn alone.

There are two other things you could blame for the war that would have made a much larger difference had they not happened this way.

If Jon Arryn had not been poisoned, Ned wouldn't have been brought south thus Robert very likely would have lived a lot longer. No dead king, no honorable Hand to stir shit up, not succession war. Following this theory, you could sorta blame the war on Lysa Arryn and Littlefinger, since they were the ones behind the poisoning.

You could also go back farther and point the finger at Cersei and Jaime. If there was no incest going on and no question of the father of the heir (by way of it really being Robert's and having the dark hair and such), there would have been no grounds whatsoever for a succession war and the idea probably never would have gotten into Stannis or Renly's head, and even if things still happened the same otherwise Ned would have fully supported Robert's son taking the throne so there wouldn't have been any drama in the North.

The only moment that could truely be considered the cause of the war of five kings is the true king's death, but really the entire cause of the war was Ned's actions leading up to his death. Had Ned not decided to tell every untrustworthy person that he'd discovered the incest and failed to tell Robert the war of five kings would have not happened or at the very least would have gone differently. Had he not told Cersei she wouldn't have caused Robert's accident, had he made it a thing before Robert died then Cersei and the kids would have been fucked and at the worst it would have ended up a war of two kings with the Starks with Stannis and the Lannisters with Renly. Had he not told Roberts brothers they wouldn't have had the basis of their claims. Had Ned not been killed Rob wouldn't have tried the king in the north nonsense. There's also the possibility that if Ned hadn't told Cersei, Littlefinger would have killed him before he could stir shit up, that seems to me like Littlefinger's best option before Cersei started her scheming.

To MDK's theory: Preventing the assassin and Cat kidnapping Tyrion wouldn't have prevented the war, it might have prevented the war in the north from starting so soon, leaving Tyrion and Jaime to make the decisions in king's landing during Ned's investigation and Tywin showing up immidiately after Jof became king, leading to things being much better for the Lannisters. That whole plot line was mostly unconnected to the plot line that caused the war.

To Jorick: Changing either of the things you listed wouldn't just prevent the war of five kings plotline, they would prevent the entire series, it is best not to think in those terms.

Aegon VI is the one true King.

In the canon Jorick presented of Jon Arryn not being killed, then Robert would have killed a lot of major characters in the series and eliminated the chance of most people contending for the thrown. Cersei and the kids would die for sure, leaving Stannis well known as Roberts heir. All Stannis would need is for the Red Lady to do her shenanigans to kill Robert, in which case the only one who might try and challenge him for the throne is Renly, which seems unlikely that the Tyrells would push him into a war like that with Stannis likely having the North backing him.