I used to fundamentally hate this idea. But for some reason . . .after reading your explanation here and thinking about it, it would have undoubtedly been a great (not to mention gutsy) decision on Kishimoto's part and would have improved the story. As you know, the actual series cowered away from any instance of Naruto have to actually come to terms with the flaws in his ideals. All the villains simply succumb to the talk-no-jutsu (or were otherwise dispatched by outside forces) for this to actually ever be placed on the table. When confronted by Sasuke's multiple efforts to murder Sakura during the land of iron arc, what is Naruto's response? To simply sing kumbya and even preach nonsense about how both he and Sasuke would "die together" and "understand each other" in the afterlife. And Naruto justified this claptrap by claiming that if he couldn't even save one friend, he could not become hokage. Really? Last I checked, the 1st Hokage managed that just fine!!!
In terms of your idea, I LOVE the idea of Naruto leaving the village and believe that this should have happened right after the Pain arc. Take out the Land of Iron nonsense and instead have Naruto go on his own journey directly afterward (which would have been a result of having contemplated on his last conversation with Nagato and how he had intended to find a way to bring peace to the world). At this point, we get multiple mini arcs on the scale of the Zabuza arc with Naruto solving smaller scale problems while being exposed to the world without the lense of the leaf village limiting his worldviews. Because Sakura had basically been shafted since the Rescue Gaara arc, I think it would be appropriate to have her accompany Naruto initially. Just Naruto and Sakura since the series seldom gives them any solo time together. Throw some news friends and enemies in every now and then, but Naruto and Sakura remains the focus and both get some character development over the course of these mini-arcs (Naruto slowly matures and Sakura slowly begins realizing her feelings for Naruto; there could be one scene, albeit played up for laughs, that hint a tad of jealousy, towards the way Hinata protected Naruto from Pain). Naruto and Sakura mini-adventures last one year and they return to the leaf village only to see that it has become cold & despotic and ruled with an iron fist by Danzo. I won't get into too many details, but I think the series would ultimately end in another battle within the leaf village (i.e. no war arc), only Sasuke being the one to try to destroy it this time and Naruto finally coming to terms with what it will take to protect the village and bring about world peace.
I used to think that Naruto's POAL was the main obstacle in the way of this idea, but a good writer could use Naruto's failure to uphold this promise to their advantage. Believe it or not, aside from the stupidity of Sakura giving a fake confession to Naruto, I actually LIKED seeing Naruto and Sakura argue with each other afterwards. Not in the playful fashion we're accustomed to when Naruto does something stupid, but hashed out spirited debate. I think this alternate version of events would incorporate more of that when it came to the subject of Sasuke. I could see Sakura being the first to come to the conclusion that Sasuke might be beyond saving and Naruto and Sakura hashing it out over this. I could see Naruto and Sakura saying all kinds of HARSH things about one another and even splitting up temporarily. Ultimately, their division on the subject would be portrayed as a reflection of just how much each one loves the other (Naruto doesn't want to abandon his promise to Sakura and Sakura does not want to lose Naruto).
I feel like Naruto's ultimately takeaway from all of these events and even his failure to save Sasuke would be the method of bringing about world peace and ending the cycle of hatred boils down to one thing and one thing alone: The Future. The way to end the cycle of hatred is to ensure that the next generation does not succumb to it. Naruto's tenure as hokage would be focused on the leaf village's youth; he would erase the circumstances that allowed children to grow up become Sasuke, Obito, Orochimaru and Nagato. The next generation of ninja would be steered towards love and peace as opposed to hatred and isolation. There would be no better example of this than Naruto and Sakura's own children who they raise to the best of their efforts.
But yeah, the actual canon Naruto is a complete loser who is so inept that he can't even solve the problem of juggling multiple responsibilities even though he can literally appear in 1000 different places at once for extended periods of time.
I would have changed the Pain-Arc completely. I've already talked this up and down, but the meeting between Sasuke and Akatsuki was terrible. If we follow the plot that Obito managed to convince Sasuke to spark a hatred against Konoha, the events afterward make zero sense. Sasuke was persuaded to join Akatsuki as a party, because both parties have the same goal - the destruction of Konoha - but is then excluded from said goal. Sasuke had no benefits at all in this whole interaction. Why does he leave the act of revenge, which he is seeking, on Akatsuki to deal with KilleBee instead? But well, I don't have to pretend that I don't understand the intention of Kishimoto. It would have been impossible to justify further the relationship between Naruto and Sasuke in the constellation as it is, if Sasuke attacked the village and its people for real.
But that's what should have happened. Not Pain, but Sasuke attacking Konoha. Pain would have taken care of KillerBee, at which point the Raikage then proclaims a world emergency to end the affair with Akatsuki once and for all - as it also happened in the plot, only that you bypass the obsolete plot with Sasuke-Mising-Nin which has been irrelevant in retrospect. And meanwhile, we have Sasuke, who then attacks Konoha. In this event, you got countless interactions between Sasuke and his former comrades and friends. Some would try to stop him without killing him, like Sakura and Kakashi, who share a special bond. Others might intend to kill him but are inferior to him. You would have the Batman-Joker dilemma that someone unstoppable can't be stopped unless you kill him. Naruto then appears late at Konoha's gates and sees Sasuke's path of destruction. Houses in flames, bodies on the ground, the streets filled with screams and fear - perhaps even the death of Teuchi (Ayame then takes over the shop). Naruto finally acknowledges a fatal realization: Sasuke is too far gone, and his only rescue is death. They meet at last, eyes met, and without exchanging words, a long bitter fight ends with Naruto doing the killing blow. That would have been the most critical point in the whole story. This would not be a friend killed in battle, as was the case with Neji or Asuma, but the death of a friend by the world into which he was born. A young boy, still haunted by his past and exploited by antagonists who have brought him more despair. This would be the biggest fire inside Naruto to pursue Jiraiya's dream. A world where this would never happen again. The death of Sasuke will serve as an exemplary symbolic figure for him and something that will forever haunt him on the path to finding true peace.
You have saved the plot by that. Obito (Akatsuki) is further antagonized because he gives Sasuke the final push over the cliff. Pain can be further held as the final antagonist - the same motivations as Naruto, but with different goals and approaches, collide to a climax at the end, deciding who found the solution to save this corrupted world.
This leads to a trembling relationship between Sakura and Naruto, who still doesn't quite understand why Naruto had to kill him - when he was the one who risked everything to save him. A mix of disappointment, helplessness, anger, and hatred would then arise in their relationship. Until the dust settles and Sakura realizes how many victims have been created by the hand of Sasuke. She realizes at the mass funerals how a young boy who has lost his parents suddenly carries the same hatred and anger in his eyes as Sasuke once did. She gets the whole picture, the depth of the entire tragedy, and why Naruto had to kill him.
On the other hand, you have Kakashi losing another one close to him. He lost Obito and Rin to senseless wars that he never understood and has now lost his student for whom he was responsible. He and Sakura would be the first to know why Naruto had to do it, beyond the fact that the village had to be saved. They might also get the motivation to follow Jiraiya's dream and support Naruto on his upcoming path.
With Naruto, you would have a different case. The triumphant hero who doesn't feel like celebrating. The smiling faces and celebrations are cheering the death of Sasuke and, thus, the salvation of Konoha. The hateful words and insults Naruto endure toward his deceased friend. A nagging feeling that the people cheering him were the very people who despised and rejected him back then. The thought occurs to him whether he would have suffered the same fate if he had taken the same path back then - that the cutting edge between hero and villain, him and Sasuke, is so thin. That would be the moment when he should not care about the Hokage title and distance himself from what has always been his home. Konoha would be just one puzzle of many that needed change, and he longs to go out into the world to find the answer.