The story would actually have to be completely different if it were narrated by Jerry's mother because she has no idea what Jerry is doing when he goes to the wild bay on his own. Instead of being a story about Jerry's coming of age, it would be a story about his mother's anxiety over what he wants, if she's doing anything right, and how much freedom to give him. She sits on the safe beach, presumably worrying over him, while he goes away and works to be able to swim through the tunnel like the older boys. Without a narrator who knows Jerry's thoughts and feelings and what he does when he's alone, there would be no older boys, no tunnel, and no evidence of Jerry's major accomplishment in the story. It couldn't even be called "Through the Tunnel" because the audience would never know about the tunnel. It would have to be called "The Safe Beach" or "A Mother's Anxiety" or something that focuses on her perspective.