Well, even if the prosecution rested tomorrow, they can’t just sentence him, because he has to have the opportunity to make his defense.
I don’t think it matters much, in terms of process, that he admitted to the shooting, since he has pled not guilty and can still mount a defense against murder even if he admits to the shooting. The Zimmerman trial is a perfect example of that. It’s hard to imagine any defense this guy could come up with would be convincing, but the prosecutors have to make a solid case anyway. They’ve got to assume he will try to dispute the charges somehow when he gets his turn.