The void left by Hiruzen, the Third Hokage's absence following the Konoha Crush Arc in Naruto, created a power vacuum that needed filling as soon as possible. Kishimoto therefore wasted no time setting up the next Hokage as the elders of Konoha quickly convened, making a decision that drove the next story arc and the series' future. The elders, Hiruzen's former teammates, chose Jiraiya.
However, unwilling to be tethered and feeling like his destiny belonged elsewhere, Jiraiya's fate took a much different course. Instead of being honored by the position and embracing the acclaim of becoming Konoha's leading ninja, Jiraiya refused the job, passing it off to another of Hiruzen's pupils, Tsunade. Tsunade owned the position impressively, but Jiraiya was poised to become a top-tier leader for Konoha and the world.
Worldliness and Notoriety Could Aid Foreign Relations
Hiruzen's students, Jiraiya, Orochimaru and Tsunade, are renowned as the Sannin, marking them as legendary figures. To acquire such infamy and experience, they traveled the world, fighting and learning in various countries. While Orochimaru's path rendered him a villain around the globe and Tsunade became an infamous gambler, Jiraiya maintained a certain air of heroism.
Jiraiya's fame extends past being a ninja through his series of adult novels, the Icha Icha series. As a celebrity and famed warrior, Jiraiya's notoriety is undeniable, opening the door for connections with people in different countries. All it takes is one strong fan to persuade a more extensive group about their idols' importance, knowledge and skills. His fans would help Jiraiya establish stronger bonds with other nations because he is known as more than just a soldier.
Jiraiya Has the Iconic Image and Attitude
In the same way that Tsunade and Kakashi commanded respect from various world leaders, even higher regard would have been given to Jiraiya. His penchant for humor and harmless troublemaking adds charm to his well-established battle record, giving him a balanced image. Where the first two Hokages garnered less savory reputations, Jiraiya could have helped redevelop relationships with his joyful attitude.
Even the Third Hokage, with all his teacherly charm due to his active presence as a wartime leader, carried more weight as a warrior and less as a diplomat. Nevertheless, Jiraiya's penchant for humor helps bridge gaps between nations and generations. That attitude would have helped establish a brighter future supported by a larger group of people.
Tsunade Fit His Image For the Future and Potential Shadow Hokage
Even in refusing the role, Jiraiya still proved himself the proper leader of the village. Going against the Elders' plan and pushing Tsunade into the position revealed his vision for the future -- a picture of change and healing. Tsunade would be the first female Hokage and village leader specializing in recovery and battle. Jiraiya saw her as a step in the right direction.
Not intending to leave things the way he did, Jiraiya would have been able to continue his role as a shadow leader for the village. However, as Danzo did with Hiruzen, Jiraiya could have also taken care of less savory business, leaving Tsunade to shine bright and lead Konoha. The most significant difference lies in how he would approach the job. Greed, malice and ego don't drive Jiraiya, removing blinding characteristics that often sully leaders and cause people to become villains.
As an iconic figure famous for his writing and infamous for his battle prose, Jiraiya would be an immediately recognizable and respected world leader. In addition, his vibrant nature and demeanor would help alleviate tensions and recreate bonds. These factors all add up to a quality leader, with his reluctance to seal the deal cementing this fact by highlighting a lack of power-mongering. Even as a secondary, behind-the-scenes leader, Jiraiya would have drastically altered the course of Konoha for the better.