Rather than "Reach the Tournament" I would characterize our first goal as "get at least one Team Uplift member promoted to chuunin". This will be easier if some of us reach the tournament, but it's not strictly necessary; nor is reaching the Tournament sufficient. The Exam as a whole is being watched and graded all the time. In theory a sufficiently awesome performance during the other Events could get a promotion even without reaching the Tournament, where as even reaching the Tournament means nothing if you don't show chuunin-level abilities doing so.
I also think you're missing a goal. We want to create bonds with other Leaf genin and show sufficiently loyalty to Leaf and the Will of Fire to tamp down at least some of the disquiet about the team's former missing-nin status.
As for an alliance between Mist and Leaf, I think we need not go to "hardliners" for the difficulty. There is a lot of literal blood that has been spilled between the two villages. Remember, Leaf's beloved, universally respected Third Hokage literally died at the hands of Mist ninja less than three months ago. You can see why, "And now we should be friends with them," might be a hard sell. The same on the other side. No matter what their personal opinion of Yagura, all those other Mist elite left an awful lot of spouses, children, and holes in lives behind them. All of this a very, very short time ago.
Finally, I wonder whether 'minimize the number of events' is really Jiraiya's goal. Suppose he has leads on Naruto, but needs more time for them to come to fruition. The clans won't depose him in the middle of an Exam... if he feels like he just needs an extra month and that he can smooth over everything if he brings home the goods at the end of the Exams, then it might be worth the risk.
As for Looking for Naruto....
1. We have teams from many, many villages. While genin don't know much, the collective viewpoint of street level views from all the villages might reveal something interesting?
2. The streets of Mist are full of foriegn ninja, and their remaining elite have all been pressed into duty as proctors to wrangle the kids. That makes it the perfect time for an organization like Akatsuki to try... something. (Or Orochimaru for that matter.) Keep eyes open. Look, listen, think. Read each update three times and parse it for subtle clues. The QMs can't write everything that the characters see and experience, but on the other hand there's something important that the characters actually see and might realize the significance of, then it'll probably go into an update somewhere.
