There are two reasons why the author stays in the flashback. The first reason is because the story 'dies' with Doodle. Everything ends under that tree: Doodle's life, the author's innocence and our glimpse into their life. And it's a powerful ending because as the reader you're left with the image of Doodle under the tree.
That's the second reason for the story ending in the flashback. You understand fully that this is the image the author has carried with him his whole life. This knowledge makes the story more sad and delivers the 'pride can be evil' theme more powerfully.
The narrator chooses not to return to the present at the conclusion of "The Scarlet Ibis" because what occurs at the ending is the most climactic moment of the entire story.
After a childhood spent berating his young brother, Doodle, for his inadequacies and punishing him for his disabilities, the narrator is left to face the consequences of his actions: he has run far ahead of Doodle during a storm, and Doodle has died, collapsing in the mud from the great physical efforts he has taken to keep up with his older brother. We stay in this moment in the past because it carries such huge personal and emotional significance for the narrator; it is clear that this moment will remain with him for all of eternity. Flashing to the present would only kill the intensity of that truth.
Additionally, this is a deliberate narrative strategy that avoids the emotional collapse of attaching a traditional denouement to the story. The death of Doodle is shocking to the reader, and by forcing us to stay with that shock rather that giving us an easier "out" (showing the narrator growing up, laying Doodle to rest, examining his current feelings on the situation, etc.), we may experience the same burden that the narrator faces: Doodle's death will stay with us.