The only proper audit of 'lives saved' is some 20 years old and called the Sheffield study. It demonstrated that HEMS with doctor as in London saved 1-5 lives a year at a cost then equivalent to heart transplantation. This would now be more like three times a heart transplant. It failed to demonstrate lives saved for paramedic HEMS. Sadly no service has produced any peer reviewed data in the mainstream medical literature that I am aware of - happy to be proved wrong
HEMS do not task themselves. The decision to launch is taken by the ambulance service. Some services have a specific controller who oversees this. Some services will task in order to meet government targets on arrival time - so they may send a helicopter to a minor problem because they have no land based ambulances or because the land based service would take longer than the target time they have to meet. I know of one service that uses or has used midwives and first aiders to meet the targets, so if you cut your finger you may get the local butcher with a bandage or a twin engined IFR helo solely on the basis of what is available to the controller. Welcome to the logic of the NHS