Which takes....less than two minutes?
If it is that frustrating, why dont you have direct numbers stored on you cell phone? I would, and sounds like it would faster than the run around.
In my case, about ten. With someone passed out from a fall on my front walk.
Do you want to wait an extra ten minutes when someone in your house is unconscious? Ten minutes during a head injury is a long time (her's wasn't a head injury, she'd passed out from pain, but we didn't know that).
In my case, I expected 911 from my landline to work. It didn't. I think expected a dispatch direct from 911. That didn't happen either. I then expected a successful transfer from the dispatch I had called to someone who could dispatch from my city. That ALSO did not happen.
We can't dispatch an ambulance around here - or the police - from anything BUT 911. That IS the direct number for all dispatch services. If you call the police directly, they will forward you to 911.

(here's the link for anyone interested: 
doubt it....Ive been transferred before more than four times on one call and took a maximum time of about 1-1/2 minutes