What breakthroughs has Jerry achieved by the end of "Through the Tunnel"? Consider his feelings about himself, his conquest of the tunnel, and his dependence on his mother.
With the points to consider that you've listed, you've pretty much answered your own question concerning "Through the Tunnel." Jerry makes breakthroughs in all three areas you mention:
Jerry no longer has to prove himself. He's already done it. When his mother tells him at the end of the story that she doesn't think he ought to swim any more that day, she expects him to resist, but he doesn't.
She was ready for a battle of wills, but he gave in at once. It was no longer of the least importance to go to the bay.
He's already proven to himself what needs to be proven.
Jerry makes a breakthrough concerning the tunnel when he swims through it. He pays a great price for doing so (major nose bleeding), but he survives and makes it through the tunnel.
As far as his dependence on his mother, all of the responsibility it took for Jerry to train himself to swim through the tunnel is his own. His mother knew nothing about it. Of course, one could argue that she should have, and what he did was extremely dangerous for a boy his age. But as the story is, Jerry shows great intensity of purpose in his preparations for swimming through the tunnel. Any boy that can accomplish what Jerry does cannot be accused of being overly dependent on his mother.
In getting to the point where he is able to swim through the tunnel, Jerry has vastly improved his feelings about himself and he has greatly reduced his dependence on his mother.
When the story starts, he has very little self-confidence at all. You can see this in how he acts all stupid trying to make the local kids notice him. Even up to the last minute, Jerry is not really sure of himself. He is really planning not to try the swim, but all of a sudden he decides to do it.
Once he manages to swim through the tunnel, however, he is much more sure of himself. He has no need for the local boys. And he does not even have to argue with his mother.
In the story "Through the Tunnel" going through the tunnel was a task that Jerry prepared himself to do. He knew it was going to be difficult, but he had challenged himself. When he went through the tunnel he had a very hard time. It was a difficult swim and he thought about turning around but he did not. After he had reached the other side, he realized he could accomplish great things. He had gained self-confidence.
Conquering the tunnel led Jerry to feeling more independent from his mother. He had taken a step away from her apron strings. He had performed a task that she would never realize the level of difficulty. Just having this knowledge made this experience highly significant.
All three of the issues you mentioned tie in together in helping Jerry to grow up.