No, I believe you misunderstand something very important. Zimmerman is not assumed to be innocent, nor is it assumed that he is telling the truth.
Zimmerman killed an unarmed teen. He has admitted to doing so. Killing unarmed people is against the law. However, Zimmerman is claiming that he had no choice but to kill this kid, and that therefore he should be excused from the normal penalties that might apply.
Zimmerman is the one making this claim. It is his job to provide some evidence to justify his choices. The prosecution's job here is not to prove what happened, but to show that Zimmerman's version of events is not necessarily believable. Every questionable statement Zimmerman has made, every flaw in his various stories, the lack of any meaningful injuries, all these must be weighted by the jury. They are not weighting the prosecution's case -- as there is no question that Zimmerman pulled the trigger -- they are weighting Zimmerman's excuse.
The reason for this is simple. If the law did not work in this way, virtually any time a murder took place away from witnesses the defense could claim self-defense. Yes, I stopped the woman at gunpoint, but I never planned to kill her -- she grabbed for my gun! Prove me wrong! Yes, I followed this kid around with a gun, but I never planned to kill him -- he attacked me, he went for my gun! Prove me wrong!
See how that works? Zimmerman is the one making the claim, and it's a bloody big one. We have an unarmed dead teen, a teen who (by the defendant's own admission) was attempting to get away. Getting from that to "I had to kill him" is a mighty big leap, and Zimmerman's story has quite a few holes.
IMO