He is definitely under-appreciated. Even if he is used almost solely for comic relief in most tie-in material, he's actually one of the best supporting characters in TLK.
In the original movie he does come across as a pompous authority figure at first, but for the most part, he is only trying to do what is best for Simba and serve his king -- just a guy trying to do his job. He takes more abuse than practically any other character in the movie, yet he always gets back up and never stops trying. He
is somewhat pompous, but he has a razor-sharp wit as well. Yes, he does spoil Simba and Nala's fun, but he is only trying to save them from a very real threat -- heck, he actually
saves their lives when they're in the elephant graveyard. How else would have Mufasa have known where his son was?
Then, despite how much he scolded Simba for getting into trouble earlier (not to mention the way Simba teased him and used him for pouncing practice), he shows genuine sympathy for Simba before escorting Nala back to Pride Rock. The way he acts when Scar tells the pride about Simba's and Mufasa's deaths makes it clear that he really cared about them, and it's really touching how he tries to comfort the widowed Sarabi. Yet, even when things are darkest, he even makes some jokes at Scar's expense, despite the horrible circumstances he is in.
He was pretty underutilized in Simba's Pride, but that's mostly because he didn't have that much to do during the story. Timon and Pumbaa take over his cub-watching role (and do a much worse job than he does), and the only thing he actually
does is find Kiara during the fire and tell Simba that the Outsiders are on the warpath. Then when Simba sends him off to find Kiara again, we don't see him until the final scene. What a horror it must have been for Zazu to spend the night searching in vain for Kiara, only to return to Pride Rock to find it swarming with Outsiders. As if planning to get help to save Mufasa and Simba during the stampede, then getting knocked unconscious and waking up to discover that Mufasa is dead and (allegedly) so is Simba wasn't traumatic enough.
Also,
this comic. I love how Zazu is in it. He pretty much mocks Simba nearly the entire time, but he still gets him out of trouble, even helping Simba do the very thing that Zazu was mocking him for attempting.