I'd say there's no contest -- Zazu would be at least
marginally better at keeping an eye on Kiara than Timon and Pumbaa. Even though Zazu
did lose Simba and Nala, it seemed mostly due to the two of them working together to outsmart him. I'm sure Zazu would have at least a little more luck with just
one cub to watch.
On the subject of Timon and Pumbaa, though -- the way Simba keeps asking them to keep an eye on Kiara makes me wonder about his mental health. He asks them to watch her as a cub, they lose her; he asks them to watch her as she goes hunting as a young adult, they lose her; after Kovu is exiled, they either ask to or are asked to watch her (Simba straight-out orders them to do so in the SP comic), and they lose her yet again...
Isn't the definition of insanity doing the same thing over and over again and expecting a different result?

EDIT: As to why Simba never asked Zazu to watch Kiara -- since he spent most of his life with Timon and Pumbaa, he probably trusted them much more than that silly bird that he and his best friend were easily able to evade as cubs. Still, he did ask Zazu to find Kiara when the fire started (which Zazu quickly did). When he asks Zazu to find her before the battle, though, we never see how Zazu fares, and he doesn't show up again until the ceremony at the end (probably feeling very confused about why the lionesses that were out to declare war on the Pride Lands are now sitting happily on Pride Rock).