"If this is where the monarchy is headed
Count me out!
Out of service, out of Africa!
I wouldn't hang about!"
Considering that he sang that when he thought of what kind of a leader Simba might be, why didn't he leave early on in Scar's reign? Yes, Zazu likes his position, but the lyrics above indicate that conditions existed where he'd be willing to give it up. You'd have thought Scar bringing in the hyenas would have been enough to cause Zazu to hand in his resignation.
Heck, they could have had Zazu do just that and then later be the one to find Simba instead of Nala. It would have been classic irony that Zazu ends up leaving because of Scar and ends up seeking the help of SIMBA to make things right.
Also, when arriving back in the Pridelands, instead of seeing Scar attack Sarabi, we could have seen Scar trying to make Nala be his queen and Simba arrive just in time to stop him. That would have been just as dramatic, if not more, than Scar attacking Sarabi.
Of course, this would cut out "Can You Feel the Love Tonight?" and a lot of Nala's role in the movie, but it's doable.
Seriously, why did Zazu stay in the Pridelands? He's a bird. Scar can't capture him if he doesn't want to be captured.