I’d say a big part of why this doesn’t happen is because of the enormous difference between how deaths are investigated in the UK versus the United States. Local medical examiners or coroners may investigate deaths in the U.S., but in conjunction with the police as well as other agencies. However local authorities have wholly local jurisdiction, a local coroner doesn’t get to look into the death of someone who died in Iraq just because that person happened to be an American citizen. The whole concept would not make sense in our legal system.
I think the brass (and rightfully so) fears that when you let any old yokel like many local coroners essentially have free reign to investigate the military you’ll run into people who want to go on ludicrous fishing expeditions with political motivations. Imagine if some Democrat coroner in Washington State had the authority to look into military deaths and to require whomever he wished as a witness. Some coroners are just locally elected types who could be extremely partisan. Whereas others might be professional medical experts (holding an M.D.) and be appointed in a relatively non-partisan manner.
Furthermore, the United States has always affirmed that we are the ones who are going to deal with what our soldiers do on the battlefield. If they commit a crime, the crime will be investigated and punished by us with our military court system. If they are in a FF incident, it will be investigated by us and if evidence exists that they acted inappropriately, again, that will be dealt with by US. I think this is a good policy, we should never let other countries investigate our soldiers, require our soldiers to testify in such cases, or anything else. The only time we should ever allow our soldiers to be subject to such things is if they are accused of committing a crime on the soil of the country in question (for example if a soldier killed someone in a pub in the UK, then it would okay for him to be prosecuted or investigated by the UK.)
While we do sometimes assert privileges for ourselves that we do not extend to others, this isn’t an example of it. The United States isn’t in the business of trying to hold investigations of British personnel and never has been. It’s always been the Europeans who believe in moronic concepts like “universal jurisdiction” and trying to investigate or criminally prosecute American personnel for acts committed outside their national borders. Nor does the United States have coroners issuing requests to British personnel to give them testimony (of course the whole concept of the coroners inquest is far different here, while some coroners retain the power to issue subpoenas or even impanel a jury, most do not do this and typically many coroners are medical professionals who mostly handle the scientific aspects of a death.)