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.