What if...Jiraiya were made Hokage instead of Tsunade
Hi, I have to go to work soon, so I'll just get this started. This is a what if thread, not some Tsunade vs. Jiraiya thread. Please don't say he or she would make a better Hokage just to start some debate.
The purpose is to see the different kinds of alternate histories that would've taken place. Imagine this as a parallel universe.
Originally Jiraiya was offered the position of 6th Hokage immediately after the 3rd Hokage's death by Orochimaru. He turned down this position and sought out Tsunade instead.
1.) If Jiraiya had become Hokage and trained Naruto during the time-skip in Konoha, would Naruto have gone on a rampage in 4-tails form? Would Kakashi have influenced the training with the shadow clone training method? Could the 4-tails have been prevented by Yamato with Jiraiya. Essentially, Jiraiya would not have had to teach Naruto only by himself. Thus, his resources would not have been as limited for ninjutsu, genjutsu defense, and taijutsu.
1b.) However, I believe it's possible that Naruto may not have learned what could be the most important teachings of Jiraiya that have lead to Naruto becoming the destined child. Recently Nagato and Naruto have discussed the answer to how to establish "peace" in the ninja world and system. Naruto was heavily influenced by these discussions, which I'm not certain would've taken place in Konoha, or to the extent they did away from major cities were people can think with purity and without distraction. Obviously, Jiraiya wouldn't have been distracted by the Hokage duties.
1c.) In summary, I believe it's possible that Naruto's ninja abilities may have progressed further, but his true responsibilities and worldly knowledge may have taken a severe retardation.
2.) Knowledge of Pein. If Tsunade would've fought Pein without any knowledge of him, she likely would've lost as had everyone before her, including the alternate world in which Jiraiya fought. Therefor when Pein attacked Konoha, many ninja that survived because of her healing through Katsuyu would've died. I'm also uncertain as to how much knowledge Katsuyu would've returned. Jiraiya knew Pein personally, Tsunade didn't, which is why I believe that the statement, "The real one is not there." (approximately) wouldn't have been conveyed to Konoha.
Further Jiraiya has acted as a spy getting recon on Akatsuki.
2b.) Further, during Pein's attack, Katsuyu has been coordinating information across the fights to different ninja that the toads could not because of Katsuyu's ability to divide.
I have to go, and haven't addressed Jiraiya defending Konoha. Sorry it's incomplete, but it's enough to start on if you want to add or argue against.