Not desired by whom? I think transparency is desired by citizens/customers.
Do you think there are (good) reasons why obscurity is preferred?
However that exists already, without a way to track the rescue team sent to you...
"Ambulance chaser" is a rather derogatory phrase for a reason.
An emergency dispatcher could send a Text message back with a link to a private, case-limited, web page with an ETA + a map + the ambulance location in real time.
See? no "real-time public broadcast of all the addresses a medical event has occurred at".