Are 911 company dispatchers in the business of grilling callers for self-incriminating statements now?
Yes.
You must say as little as possible. Stick to a limited script which you have been trained to use, and stop talking once you have run out the script.
Nobody from the Government is your friend after an incident. That includes EMT, Firemen, Nurses, and Doctors by the way. They can be subpoenaed to testify against you.
They really aren't kidding when they tell you "Anything you say may be used against you in a court of law".