The only thing a dispatcher is powerless to do is arrest you, and that's actually by law.
Nonsense. Dispatchers - in almost every jurisdiction - are civilians trained in handling such calls and performing dispatch duties. Union rules and contracts very often prevent even injured cops from booking time as dispatchers, as those are not law enforcement positions.
I know people in this line of work in half a dozen local jurisdictions surrounding a major metro area. In no cases do dispatch employees have any legal authority of any kind, and cannot direct people to do anything - medical or otherwise. In fact they are required to ask people if they would do this or that to help an injured person, and always say "please" because the are, essentially, asking a favor of the caller. They can't make them provide CPR (or anything else) and people who can't or won't do what a dispatch employee suggests face no consequences because they absolutely are not acting with legal authority of any kind.
All of which you know, so it's kind of mysterious that you're pretending it's otherwise, unless your understanding is limited to your own county or city where they perhaps don't do what they do most everywhere else. Regardless, the Martin/Zimmerman case does not involve a copy giving orders to Zimmerman. The dispatcher said, "OK. We don't need you to do that." And testified that it wasn't an order, meant to me one, or coming from anyone who could give one.