One obvious reason why the HEMS accident rate in the US is so high is because there are just so many flights.
The threshold for calling a helicopter to the scene of an accident is pretty low in many jurisdictions. First responders, concerned about lawsuits, treat every injury -minor or otherwise- as if it could be potentially life-threatening. Hospital doctors, too, are under pressure to err on the side of caution, ordering expensive treatment (to include helicopter transfers) that often goes beyond the actual needs of the patient. For-profit helicopter operators are only too happy to oblige.
Many studies have examined the effect of helicopter transfers on patient outcomes, and there clearly is a benefit. But it is a case of diminishing returns, with the benefit of transporting a patient with, say, a broken ankle approaching zero. Simply flying less will go a long way towards reducing the accident rate by reducing the exposure to risk. Unfortunately, it would require a wholesale overhaul, not just of the US EMS helicopter industry but of the entire healthcare system and the legal system to accomplish such a reduction on a nationwide scale.