Regardless of who calls for an ambulance the patient receiving the service is responsible for paying the bill.
+1 to this... I have Medibank Private Extras cover, (no hospital) and they must hate me right now. I broke my ankle late last year, I was alone at home in a 3rd story apartment and I had no choice but to call an Ambulance. The bill was $822 from Subiaco to Charlies *cough cough*... also had about $600 worth of dental work done last year... the whole policy costs me $120 per year.
Ambulance cover alone is available for about $40 a year... you'd me mad not to have it because you never know when you are going to need it.
You aren't responsible for the Police's actions, and I think they did the right thing - although it depends on the situation. As you know, if they have a concern for someone's welfare, they have to treat it as a high priority regardless of which emergency services are involved.
You did the right thing.