Exactly. If the Prosecution didn't call the cops and such as fact witnesses, the Defense Attorneys would have done so, elicited the same favorable testimony, and slammed the prosecutors for "hiding" them from the jury.
One way for the Prosecution to deal with unhelpful facts is to bring them out themselves, on the theory that the jury will hear them anyway, and it is better to hear them first from the Prosecution.
Of course that presupposes that the unhelpful facts are few, and do not affirmatively disprove your case.
Any testimony by cops about anything Zimmerman might have said could only be admitted if the prosecution asked for it or let the defense do so, or Zimmerman agreed to take the stand. Had the prosecutor not asked such questions, the defense could not have done so without allowing the prosecutor to cross-examine Zimmerman.