If a third person had narrated Douglas' experience, the impact of the story would have lost the reader's deep connection with the main protagonist and his fear of water. The narrator then would be passively telling the story from the perspective of an observer. The incident of drowning in water could never have successfully communicated the feeling of the "stark terror" that Douglas underwent.
In third person narrative, the 8th and 9th paragraph of the story would be as follows:
"He flailed at the surface of the water, swallowed and choked. He tried to bring his legs up but they hung as dead weights, paralyzed and rigid. A great force was pulling him under. He screamed, but only the water heard him. He had started on the long journey back to the bottom of the pool."
"He struck at the water as he went down; expending his strength as one in a nightmare, fights an irresistible force. He had lost all his breath. His lungs ached. His head throbbed. He was getting dizzy. But he remembered the strategy - he would spring from the bottom of the pool and come like a cork to the surface. He would lie flat on the water, strike out with his arms, and thrash with his legs. Then he would get to the edge of the pool and be safe."
So, it is only the first person narrative that keeps the reader gripped to the story. It makes the experience more relevant and tangible for the reader. It engages him by making him go through the experience along with the protagonist. The desperation and helplessness of being in water, which has almost become fatal, the mental and physical agony of trying to survive the crisis, the long struggle of overcoming the fear bit-by-bit and the jubilation of conquering it at the end; all make the reader feel part of the experience. The first person narrative makes the story a fast-paced and urgent reading for the readers. All this would have been lost had it been a third person narrative or from the point of view of an observer.