There are controlled studies on secondary transfers. They reduce mortality by 50%. Primary missions suffer from very little published data but what there is suggests a system crewed by doctors prevent death in serious head injuries only. So it is true they don't in general save lives.
It is not the same as being a surgeon. As a surgeon I can do anything within the limits of my equipment without any consideration except for my patient. As a pilot the need is for me to not use my equipment because of the addition of weather.
When I started in the US I actually had an experienced Nam pilot respond to my flight request with 'Nope not with this weather......is it serious?'
As a result my system has always operated a Chinese wall where the pilots don't know the medical indication and the medical crew don't know the met. And if one of my doctors even jokingly criticised a pilot he would be fired on the spot.
Sadly in the US some doctors still believe they are god although in fairness the driving force for helicopters is not the medical profession but the finance directors of the hospitals who need the ability to get high revenue patients in adequate numbers to fund regional services