I think the only thing that actually improved with use and training is the Rasengan, and even that didn't require any training to gain initially.
But
He trained for a few months and became much better than at least one of them who practiced it all his life (Jiraya)
He trained for several weeks to accomplish it. We don't know if this unusual,
To be fair, he does train a lot at being nice - at least he practices it all the time.
Now of course he also knows the basic techniques that all ninjas know, and he does train the acquire and to improve them.
You're not really arguing that he doesn't train except for the Rasengan. You're arguing that he does train, but that training works better for him than for other people because of his unearned advantages, so his training doesn't really count
But everything you train in works that way. I know how to program a computer. I had to train to do so, but there are other people who would have to train much longer to do the same thing and still other people who could not do it no matter how much training they did. That doesn't mean I didn't train, or that my ability to do it is entirely due to luck, even though it's certainly partly due to luck.
All the things that make him powerful, important, interesting, and not-dead-long-ago are plot armor and unearned gifts. Take all of them away and he literally wouldn't have made it out of the Ninja Academy.
And if you take away my computer programming potential--a potential which not every person has--I literally wouldn't have made it out of a real-life academy. I still trained in it.
You're not really arguing that he doesn't train except for the Rasengan. You're arguing that he does train, but that training works better for him than for other people because of his unearned advantages, so his training doesn't really count
No. I'm arguing that the things he trains at are not the things that make him successful. Even if he trained much less, he would still achieve all the same outcomes due to his unique unearned untrained powers. He succeeds because he's a privileged magical shonen protagonist, not because he's training rigorously.
As people who care about rationality and winning, it's pretty important to care about training. Repeated practice is how humans acquire skills, and skills are what we use for winning.
Unfortunately, it's sometimes hard to get System 1 fully on board with the fact that repeated, difficult, sometimes tedious practice is how we become awesome. I find fiction to be one of the most useful ways of communicating things like this to my S1. It would be great to have a repository of fiction that shows characters practicing skills, mastering them, and becoming awesome, to help this really sink in.
However, in fiction the following tropes are a lot more common:
Example of exactly the wrong thing:
The Hunger Games - Katniss is explicitly up against the Pledges who have trained their whole lives for this one thing, but she has … something special that causes her to win. Also archery is her greatest skill, and she's already awesome at it from the beginning of the story and never spends time practicing.
Close-but-not-perfect examples of the right thing:
The Pillars of the Earth - Jack pretty explicitly has to travel around Europe to acquire the skills he needs to become great. Much of the practice is off-screen, but it's at least a pretty significant part of the journey.
The Honor Harrington series: the books depict Honor, as well as the people around her, rising through the ranks of the military and gradually levelling up, with emphasis on dedication to training, and that training is often depicted onscreen – but the skills she's training in herself and her subordinates aren't nearly as relevant as the "tactical genius" that she seems to have been born with.
I'd like to put out a request for fiction that has this quality. I'll also take examples of fiction that fails badly at this quality, to add to the list of examples, or of TVTropes keywords that would be useful to mine. Internet hivemind, help?