I doubt there is any job in the world, where mistakes don't happen. You have doctors that spend years in school, only to perform a surgery and make a mistake. Happens all the time. Why do some people hate the cops and dispatchers? No idea. Of course, the people who make certain comments, I'm sure have never set foot in a dispatch center or have taken the wide variety of calls that are phoned in.
I suppose its like the martial arts. You get some that don't like a particular art, and thrive on making it their life to tell everyone that X art sucks. Same thing with this situation.
The dispatcher is the first person on the scene of any situation, so to speak, and they're also people that are shown alot of disrespect. In this job, you need to have a thick skin. Am I perfect? Far from it, but I attempt to perform my job to the best of my ability. The people I work with fortunately all look out for one another. If someone is not doing something right, doesnt realize the mistake and someone else notices this, they'll speak up, rather than let the person sink.
Like I said, there was an obvious mistake made here, and it needs to be looked at. Would the caller be alive today if the dispatcher had sent help? Unknown, because even if help was sent, it still could have been too late.
Its a stressful job, and certainly not for everyone. As I said, nobody is perfect.