Disclaimer: I don't own this!
Zuko never wanted to tell people that the thrill of hunting the Avatar, was mocking the water tribe peasant girl. It was thrilling to chase her, and then hear her voice again.
But if you must know, Zuko would never admit to anyone that the thrill of hunting the Avatar was because of her.
Without her, Zuko would have nothing gained at all. So why not prolong his destiny, just to see her face again for one more time.